Eddy Mach wave noise from a simplified model of a supersonic mixing layer

October,

1969.

EDDY MACH WAVE NorSE FROM A

SIMPLIFIED MODEL

OF

A SUPERSONIC

MIXING

LAYER

by

H.

S. Ribner

EDDY MACH WAVE NOISE FROM A

SIMPLIFIED MODEL OF A SUPERSONIC

MIXING LAYER

by

H

.

S. Ribner

Manuscript received Ju1y 1

9

6

9

ACKNOWLEDGEMENT

This research

was sponsored

by

the Air Force Office of Scientific

Research, Office of Aerospace Research, United States Air Force, under

AFOSR Grant No.

67-o672A.

SUMMARY

A simplified flow model is presented for simulating features of

noise generation by supersonic jets and rockets.

'Eddy Mach waves' appear

as a consequence of balancing internal and external pressures, and noise

power may be estimated.

Specifically, the turbulent IDlxlng layer of a supersonic jet is

modeled as a layer of two-dimensional square 'eddies'; this separates the

main flow U. from fluid at rest and moves at an intermediate speed.

(In

an improvedJmodel the square eddies are replaced by a turbulent flow layer

constrained by Plans. interfaces).

The initially plane interfaces, because

of unopposed internal pressures, will ripple slightly such that Mach waves

a

rise and effect a pressure balance

.

The Mach wave noise pattern is

readily calculated as well as the acoustic energy flux.

In an example simulating a round rocket jet the effective mixing

layer area is taken to be

6

~ (diameters)2, the r.m.s. eddy velocity is

0.1 Uj' and Uj

= 6

times external sound speed.

The efficiency (noise power

/

flow power) comes out to be about 1/3%, which is of the order of measured

values for rocket noise.

TABLE OF CONTENTS

NOTATION

INTRODUCTION

THE MODEL

RIPPLE AMPLITUDE

ACOUSTIC ENERGY FLOW

NOISE FROM ROuND JET

EXAMPLE AND DISCUSSION

IMPROVED FLOW MODEL

COMPARISON WITH OTHER MODELS

REFER ENC ES

FIGURES

'.

1

1

2

3

3

4

4

6

7

c

D

f

I

k

L

M

n

p

U

u

,

v

x,

y

w

13

8

'I/J

p 1]

Subscripts

abs

j

m

0 "

NOTATION

sound speed

jet diameter

frequency of eddy Mach waves

acoustic ene

r

gy

flux

per unit area normal to

mixing layer

wave nurnber

(2H/wave length)

edge length of square cells ('eddies')

Mach

number

based on external sound speed (U/c )

o

effective length of mixing region (diameters)

Perturbation pressure

(p

_abs

-p)

₀

local mean flow speed

components of local

perturbation

velocity

co-ordinate

frame (x aligned with flow)

r.m.s

.

(spatial

average) velocity

in square 'eddies'

.J

_M

2

-1

m

angle with

horizontal of

normal

to Mach

waves

stream

function

density

efficiency (noise power/flow power)

absolute

main jet flow

mixing layer

ambient (outside

jet)

INTRODUCT ION

The present paper studies a s

i

mplified flow model simulating features

of 'eddy Mach wave' generatio

n

i

n

t

h

e

n

oise from supersonic jets and rockets

.

Shadowgraphs show these Mach waves emanat

i

ng from the turbulent mixing layer

s

eparating the core of the

j

et from the

fu~bient

fluid. This mixing layer

is not of constant width, b

u

t

c

on

t

i

nu

a

ll

y sp

r

eads in the downstream direction.

It can be described loosely i

n

te

r

ms i

rr

egular unsteady 'eddies' which defy

a precise description.

In this model the ir

r

eg

u

larity and the spreading are suppressed in

a deliberate o

v

e

r

simplification

.

The mi

x

ing layer is postulated as an array

of square flow cells (models of eddies) convected at the supersonic mean

flow speed of the layer

.

It is thought tha

t

some broad features of the

phy-sical processes responsible for noise ge

n

e

r

ation are retained and displayed

ve

r

y simply

.

THE MODEL

The

s

quare eddy model o

f

the mi

xin

g laye

r

i

s

sho

w

n in Fig.l

.

The

eddy flow i

s

given by

w

c

o

s

kx cos ky

u

_J2

( 1)

w

si

n kx

sin

ky

v

=

'J2'"

with

s

tre

a

m fun

c

tion

?jJ

w

c

o

s kx

s

in

k

y

kJ2

( 2)

He

r

e w i

s

the r

.

m

.

s

.

velo

city in t

he

c

e

ll

a

n

d the wave number k is related

to the laye

r

thickne

ss

L b

y kL

=

rr

.

(

Th

e laye

r

may be general

i

zed to n cells

thicknes

s i

f des

i

red u

si

ng

kL

=

nrr~

t

he ph

ysic

al arguments will be unaltered).

It is a

ss

umed that w i

s suffi

c

i

e

nt

l

y s

ma

l

l so that the eddy flow (not the

main flow) may be

c

onside

r

ed

inc

o

m

p

r

e

ss

ible

f

o

r

simplicity

.

The pressure

field in t

h

is laye

r

would be

, if c

o

n

st

r

a

in

ed between rigid plane boundaries

ky

=

0, - rr

s

o t

h

at (1) ho

l

d

s accur

a

t

el

y

,

2

p

w

m.

_(co

_{s 2}

_k

_y

_-

_c

_o

_{s 2}

_kx

₎

+

constant

2

by integ

r

ation of Euler's equa

ti

o

n

s

.

T

he a

v

e

r

age value of this along the

boundarie

s

ky

=

0, - rr is taken to

m

a

t

ch the ambient pressure p . ,The

o

perturbation pressure p b

_{a s o}

- P

is th

e

n

2

p

p

w

_

m

""'

2

- -

cos

2 kx

(4)

The boundaries, however, are not rigid and are not capable of

sup-porting the pressure difference p of equation

(4).

On removing the artificial

constraint of rigidity the boundaries

will

deform into ripples which, because

of their downstream motion, will generate Mach waves (Fig.2). An equilibrium

can be specified in which the Mach

wave

pressure field will balance the 'eddy'

pressure field

(4).

It will be justified a posteriori that the distortion will be so

small that the errors introduced into equations (1) to

(4)

will be negligible.

Then the pressure enforced in the wave field is merely (4) at the interface

y

=

0,

and for y

~

0

it is

p

-2

P

w

m

2

cos 2k (x

+

f3y)

which corresponds to Mach waves inclined as dictated by the Mach

,

number M

=

UJc (Ref

.1).

m

Equations

(1)

to (5) apply in a frame of reference moving in the

positive x-direction with the ripple speed Um.

In a stàtionary frame (5)

becomes

p

2

P

w

m

- 2 -

cos

2

kx

(x - U t

_m

+

f3y)

(6)

which exhibits the Mach waves as moving sound waves.

The waves move past a

fixed point with speed Um' giving rise to frequency f

=

k

uJ~

=

UJL.

The sound pressure

(5)

or

(6)

in this Mach wave field by our

physi-cal argument matches the 'eddy' pressure field at the boundary of the mixing

layer; we may call the latter the 'near fi

'

eld' of the simulated turbule nt

jet. The Mach waves thus extend the near field pressures to the radiation

field.

RIPPLE AMPLITUDE

We now explore whether the

ripple

amplitude is small compared with

the 'eddy' edge length L as has been assumed.

The boundary conditions are:

a) The pressure must be equal

011

opposite sides of the rippled

inter-face.

This is met by

(5)

and

(6).

b) The streamline slopes must be equal on opposite sides, in the

ripple-attached

reference

frame.

Condition (b) will be

automatically

enforced when we choose the

ripple

u

amplitude compatible with the Mach wave pattern

(5);

it is given by

(Ref.l)

.

[

p

f3

U 2

Po m

o

2

2

1

Pm

(

~m)

t3

COS

2kx

2"

Po

2

1

Pm

(~

)

t3

sin 2kx

y

=

_4k

-Po

_m

(8 )

Since

kL

TT

the

ripple

ampli

tu

de

is

2

t3

_{Pm (w )}

_L

Ymax

_Tm

_Po

_Urn

in terms of the r.m.s. turbulence level

w/U.

As a severe test case

take~

w/U

m

= 1/3 (an upper limit),

M

m

3

(or

t3

=m

2

~), prnfp

o

= 1. Then

Ymax

= 0.025

L

(10 )

This confirms that the

ripple~

am

plitude is very much less than the

eddy length

L.

The distortio

n

of

the initial square shape

may thereby be

neglected insofar

as

its effect on

the internal

pressure and velocity fields

is concerned. This neglect is

not wholly

defensible at the higher turbulence

velocities

where

the assumption of

incompressible internal

flow becomes poor.

ACOUSTIC ENERGY FLOW

The

flux

of acoustic e

n

ergy

through

unit area is p2/p c

in a

direction normal to the

Mach waves.

The vertical

component in

0 0

Fig. 2 is

2"

I

-

p

----

si

n

8

Poco

The mean square pressure

is P

2w

4

/8

fr

o

m

(6),

and sin8

=

~

2_1/M

t3

/

M,

m m m

m

giving

I

2

4

Pm

w

8

P

o

(n)

as the flux of acoustic ene

r

gy

through unit

area normal to the mixing layer.

NOISE FROM ROUND JET

We

will

curve

the

plane

mlXlng

layer, equations (1) to

(3),

into

a cylinder to simulate the mixing layer of a round jet (Fig.3).

Since our

model is

at

best a crude one we

shall

assume that the energy flux per unit

area, equation (11), is still app

r

oxi

m

ately valid.

The effective

length

of the

supersonic

mlXlng layer will be taken

as n diameters, giving a

total surface area

nD.

no.

The

acoust

ic

power

radiated from this

area will

then be

noise

power

= TT

n

D I

2

(1

2)

And

by

comparison the mechanical power expended in the kinetic energy of

the jet flow will be

flow

power

8

P.U.

_{J J}

3

(13 )

The ratio (noise power/flow power) is the

acous

tical

efficiency

2

4

2n

_:_m_

p

_._

(U

~

)

o J

J

(1

4

)

where

the mean speed Urn of the mixing layer has been taken to

be (1

/2

)

U.'

J

EXAMPLE

AND

DISCUSSION

In an example

intended

to simulate

crudely

a

rocket jet

-it

will

be convenient to

a

pproximate

p

as a

geometric mean density.

The

m

example

specif

ic

ations are:

n

=

6,

U.

= 6

c

.

w

J O '

2

0.1 U. ;

p

/p

p.

=

1

J

m

0

J

Insertion into

(14)

yields the acoustical efficiency as

~

=

0.0034

or

0.34%

(15)

It is rather striking that this is of the order of magnitude of measured

efficiences of rocket jet noise generation.

Other proposed mechanisms for the noise generation - for example

the Lighthill quadrupole model with supersonic convection factor

-

do

pre-dict comparable efficiences (Refs.

2

-

4).

Thus the present model

cannot

be

affirmed as

necessarily

simulating

a dominant mechanism of noise

genera-tion by supersonic jets. The physical arguments

with

support of the

example

results

sugges

t

it may be

an

important mechanism.

A word about accuracy: the equations employed herein, as applied

to the specified flow model, are as

accurate

as the

Lighthill

equations

for

flow noise

in

their general form, and are more

accura

te than the usual

solution

with the

const

ant

density approximation.

The approximation

is

in

the flow model, not in the mathematics

.

IMPROVED

FLOW MODEL

It is possible to improve the model to better simulate features

of

a turbulent mixing layer

while

still retaining the

basic

simplici ty.

In the

improved model

we

replace the square eddies by random turbulence, retaining

the plane interfaces. To show that this is possible consider tÈat

a

infinite uniform flow with superposed turbulence and its mirror image will

be separated by aplane stream surface.

In

the model we remove the image

'

flow and

replace

it by fluid at

rest.

As in the square-eddy model, unopposeà

pressures on

the

plane interface

will cause

it to

ripple

and develop Mach

waves

to effect a pressure balance.

The

pressure field, either in the simulated layer, or in the Mach

wave field is no longer sinusoidal, but is random.

If

we

make

a

Fourier

analysis,

the square

eddy solution behaves

like

a single sinusoidal Fourier

component

of this more general patte

rn

.

The Mach wave

pressure field will

n

ow

have

a broad band spectrum

in

place of

the former

single line or pure

tone.

However, the results for

mean

square pressure and noise-generating

efficiency will be

unaltered, assuming

postulated turbulence is

a

'frozen'

co

nv

ected pattern

.

The improved model still differs

c

_{from a real turbulent mixing}

layer in retaining

quasi-planar

interfaces. Nevertheless,

it

wou~d s~em

that

a mechanism

rather

like

the

one proposed

here

is operative in the

~eal

flow

to

generate

'eddy

Mach

waves'.

Their role

is

to match

the

near field

pres

sur

es and extend them to

the radiation

field. The virtue of the model

is the conceptual

simplicity of

the

generation p

r

ocess

and

the ease with

which

predictive

formulas can

be ext

r

acted

.

COMPARISON

WITH

OT

HER M

ODELS

The Lighthill

model as extended by

Ff

owcs Williams (Ref

.2

) or by

Ri

b

n

e

r

(Ref.3)

deals

with

a

supersonic

mixi

n

g layer in

terms

o~

the

source-motion Doppler

e

f

fect

.

This

is

embodied

in a

'convection factor' that

multiplies the

int

ensity

that would

occur

for unconvected

eddy sound sources.

The

emission has

the

characte

r

of

Mach waves, with

peak

intensity

normal

to

the Mach cones

of

the

moving eddie

s.

The ed

dy

sound

sources

are identified

as perturbations of tne

Reyn

olds

stresses

or

r

ate of

momentum flow.

These

generate a near field

(

'p

se\ld.

osoun

d

'

)

a

nd.

a far field

.

The convection factor

dictates the 'eddy

Mach wave' character

of the

far field,

but

it does

not

apply

to the near

fiel

d

,

since the

moving ed

di

es emi

t

into

locally co-movi

n

g fluid.

Thus the

extended

Li

ght

hi

l

l

theory

predicts both the near field

and

the

far

field

in

terms o

f

pe

rturbati

ons of

rat

e of momentum flow in

the

~ddies

together with

eddy

c

onvection effe

ct

s

.

An integral formalism is

us

ed

,

and it

is

as

ac

cur

ate as the

turbulence description

fed into it.

A

n

other important model

is that

of Phil

li

ps (Refs. 3,5). His eddy

sound

sources - equivalent

to Lighthill's in their

tot al emission - are

identified as products of pairs of

velocity

gradients

.

(These can be

~elated

t

o

t

he deformation of fluid elements).

The

method of solution applied to a

supersonic shear

layer leads

directly

to

~ddy

Mach waves as the far field;

.

'

the

near

field

is not

obtained although

in

p

r

inciple it could beo

.

' .

In the

dilatation theo

ry

(Refs.

3,6) the

eddy sound sour

.

ces are

identified

as

the

volume fluctuations of

fluid

elements in response to the

near field

pressure

.

This

leads

to

an

int

egral

re

lat

i:ç.g

. t

.

he far field

pressure to

(essentially) the near field pressure .

In tl1is reppeet it

<

.

resembles the

present

model (although sharing the

ri

gor of

the

Lighthill

formalism) , but in the present model the connection is much more

direct:

a matching,

in facto

The present

model,

lik

e

the

othe

rs~

predicts

both near field

and

far field sound

f

r

om

the 'turbulence

'

flow

fie

ld

properties.

But the

turbulence

is

highly idealized, and

th

e

continu

ous shear in the mixing

layer is idealized into two

plana

r

slip

s

urfac

es

.

The greater simplici ty

appears in both the physics and mathematics of

th

e

Mach wave generàtiQn

p

rocess .

.

.,.

1. Liepmann, H. W.

Roshko, A.

2. Lighthill,

M •• J.

3. Ribner, H. S.

4. Ffowes Williams,

J.

E.

5. Phillips, O. M.

6. Ribner, H. S.

REFERENCES

Element

s

of Gasdynamies, John Wiley

&

Sons,

New York 1957, (Lib. Congress 56-9823), p. 426.

"Wright Brothers Leeture. Jet Noise" AlM

Journal

.

Vol.l, no.?, pp.1507-1517, July,

196

3

.

"The Generation of Sound by Turbulent Jets" ,

Vol.8. Advanees in Applied Meehanies,

pp.1

03

-182, Academie Press, New York (1964).

"Some Open Q.uestions on the Jet Noise Problem"

Unpublished paper (1967). Dept. of Mathematies,

lmperial University.

"On the Generation of Sound by Supersonie

Turbulent Shear L

a

yers ,

J.

Fluid Meeh. 9,

1-

2

8 (1

9

6

0

).

-"Aerodyn

a

mie Sound from Fluid Dilatations.

A Theory of the

S

ound fr

o

m Jets and Other

Flo

ws

" .

Uni v

.

of Toronto, lnst. Aerosp

i:

lCe

Studies, UTIAS Report. 86, (1

9

62).

·

Ky

0

_{FLUID AT REST}

0h\

l'S

I~

KX

~

MIXING

~

U

_m

LAYER

~

I

l}---/

I '---../

'---../

-1T

To

MAIN FLOW

~

U·

_J

FIG. I MODEL SIMULATING FEATURES OF ROCKET NOISE GENERATION .

I. TURBULENT MIXING LAYER IS SIMULATED BY AN ARRAY OF SQUARE

"EDDIES" . NOTICE UNBALANCED PRESSURES ACROSS PLANE INTERFACES.

EDDY MACH WAVFS

,I"I~I~

---""'.~

Urn

MOVING RIPPLES IN STATIONARY FRAME

P2

/pë

FIG.2

MODEL SIMULATING FEATURES OF ROCKET NOISE GENERATION .

11. TRAVELLING RIPPLES DEVELOP IN INTERFACFS AND .GENERATE 'EDDY

MACH WAVES· (NOISE) ; WAVE AMPLITUDE IS SUCH AS TO ACHlEVE A

PRESSURE BALANCE ACROSS INTERFACES.

~~~I~~

R~:D

I .

nD

SUPERSONIC ZONE

FIG.3 MODEL SIMULATING FEATURES OF ROCKET NOISE

GENERATION . 111. GEOMETRY FOR ESTIMATING RATIO (NOISE

POWER

IJ

ET FLOW POWER) FOR SIMPLIFIED "ROCKET JET".

UN'CtASS rF rE!)

Security Classification

DOCUMENT COHTROL DATA· R&D

(Securily cl •• ait/cal/on ol tIlIe. body of abslract and Indeaih/l _notal/on mual be entered ""'an !he overelI report la cf. •• il/ed) 1. O~IGINA

TIN

G

ACTlvl,Y

(Corpora te aulhor) 2a. ~EPORT

SECURI TY C LASSIFICA TlON

University of Toronto.,

UNCLASSIFIED

Institute for Aerospace Studies,

Zb.

GROUP

Toronto 5 , Ontario Canada.

3.

REPORT TlTLE

EDDY MACH WAVE NOISE FROM A SIMPLIFIED MODEL OF A S UPERSONIC MIXING LAYER

4·

DESCRIPTIVE NOTES

(Type ol report anct In"lualve dat ... )

Scientific

Interim

5·

AUTHOR(S)

(Laat name. lirat name. In/llat)

H. S. Ribner

6·

REPORT DATE

7a·

TOTAL NO. OF PAGES

1

7b.

NO.

6

0F REFS

October, 1969.

7

Sa.

CONTRACT OR GRANT No.AF-AFOSR-67-0672A

ga.

ORIGINATOR'S REPORT NUMBER(S)

b.

PROJECT NO.

9781-02

urIAS Technical Note No.146

,

c.

61102F

lb.

OTHER RJPORT No(S)

(Anyothernumber. thatmaybe ••• I""..,

thl. report

d.

₆₈₁₃₀₇

AFOSR 69-2145TR

10.

AVAILABILITY/LIMITATION NOTICES

l .

This document has been approved for public release and sale;

its distribution i$' unlimi ted.

11.

SUPPL EMENTARY NOTES

12· SPONSORING MILITARY ACTIVITY

A.F. Office of

Scientific

Research (SREM)

TECH,

OTHER

"4.,1;"

₁₄₀₀

_Wilson

_Blvd,

.

Arlington, Virginia 22209, USA

13.

ABSTRACT

A simplified flow model is presented for simulating features of noise generation

by supersonic jets and rockets.

'Eddy Mach

waves' appear

as a consequence of

balancing internal and external pressures, and noise power may be estimated.

Specifically, the turbulent mixing layer of a supersonic jet is modeled as a

layer of two-dimensional square 'eddies'; this

separates

_{the main flow Uj from}

fluid at rest and moves at an intermediate

speed.

(In an improved model the

square eddies are replaced by a turbulent flow layer constrained by plane

interfaces) • The initially plane interfaces, because of unopposed internal

pressures,

will

ripple

slightly

such that Mach

waves

arise and effect a pressure

halance. The Mach

wave

noise pattern is readily calculated as weIl as the

acoustic energy flux.

In an example simulating a round rocket jet the effective

mixing layer area is taken to be 6 H(diameters)2, the r.m.s. eddy velocity is

0.1 Uj , and Uj

=

6 times external sound speed. The

efficiency

(noise power/

flow power) comes out to be about 1/3%,

which

is of the order of measured values

for rocket noise •

•

"

UNCLASS !rIED

Security Clasaification

14·

KEY WORDS LIIiIK A

LINK. LINK C "OLIt WT "OLIt WT

1) Jet Noise

2) Rocket Noise

3)

Acoustics

4)

Aeroacoustics

INSTRUCTIONS I. ORIGINA'nNG ACTIVITY: Enter tbe n_e .nd addr . . .

of the contractor, .ubcOfttractOl', lP'.ntee, Department of De-fenae acUvlty Ol' otht'l' ora.nb.Uon (corporate author) l.aulnc the report.

2a. REPORT SECUJlTY CLAS8IFICATION: Enter the over-all aecurlty CI ••• Ulc,Uon of the report. Indlcate whether "Reatricted Dat." 1. lncluded. Marklhlla to b. In ac:cord-ance wUh appropri.te .ecurlty relul.Uon ..

2b. GROUP: Automatic doWftlt.cIlnc 1 •• pecUled In DoD Dl-rectlve 5200.10 end Armed Force. Indu.trlal Manual. Enter the group number. Alao, when .pplic.ble, ahow that optlonal markinga h.ve been uaed for Group 3 and Group .. a • • uthOl'-ized.

3. REPORT TlTLE: Enter the complete report tltle In all c.pital letter.. TUle. in all c . . . hould be unclaaalfled.

U a meanlnlful tlUe C.MOt be . . lected without claaallic ..

ti,)n, ahow Utle cl •• aif1catlon in all c.pital. In p .... nthe.i. immediately followinc the UU ..

4. DESCRIPTIVE NOTES: If appropriate, enter the type of report, e.g., interim, proarea., .ummary, annual, or flnal. Oive t~ inclu.lve· date. when • apecific reportina perlod la covered.

5. AtrrHOR(S): Enter the n.III8(.) of author(.) •• ahown on or in the report. . E'ltet· l.at name, flut aame, mlddle Initi.!.

If :rJlitatY, ahow r.nk .nd branch ol .... vic .. The name of the princip.1 .athor ia .n .b.olute minimum requlrement.

6. REPORT DATE.: Ent ... the date of the report a. d.y, montt!, ye.r; or month, ye.,. If more tban ooe ct.te appe.r. on the report, u.e d.te of publie.tlon.

7 •• TOTAL NUMBER OF PAGES: The tot al pqe cou~

ahould !ollow nonnal paclnatlon prOcedure., 1. .. , enter the number of p.le. cont.iD1nc lnlorm.tion.

7b. NUMBER OF REFERENCEa Ent ... tb. total numb., of referenee. clted 10 th. report.

Sa. CONTRACT OR GRANT NUMBER: If approprlate, enter the applic.ble numb., of the contr.ct or lI'ant under whlch the report w •• wrltten.

Sb, Sc, • Id. PROJECT NUMBER: Ent., the approprlat. military departm.nt ldentifie.Uo.n, auch a. project aumb." aUbproJect number, .y.tem numbera, t •• k numb.r, ete. 9a. ORIOINATOR'I RItPQRT NUMBER(8): Enter the olB-ci.1 report number by wblch the documeDt wUI ba ldenUfled .nd controll.d by th. or1ilnatinl ac:tlvity. N . number mu.t be unlque to thl. report.

9b. OTHER REPORT NUMBER(S): If th. report ha. been saai,ned .ny oth., r~rt numb • • (.Uher by th. on,lnetor or by the .,x>neor). alao enter thl. ~ber(.).

10. ,WAILABlLITY/LDIITA'nON NOTICES: Enter any U . . U.Uona on furUwr di •• em1o.Uon of th. report, othe, than tbo ••

impo.ed by aecurity cl •• atricstion, uairll .t.nd.rd .t.tementa .uch aa:

(1) "Qualified requeater. may obtsin copie. of thi. report from DDe."

(2) "Foreip .nnOURcement .nd diaaemination of thi. report by DDe i. not .uthorized."

(3) "U. S. Government .,encles may obt.in copie. of thi. report directly from DDe. Other qualified DDe uaera ahall reque.t throu,h

--

...

---

."

(4) "U. S. military .Ienciea may obt.in copie. of thi.

report directly lrom DDC Other qualified uaera .h.11 request throu,h

--

...

--

...

(5) "All di.tribuUon of thi. report i. controUed.

Qual-ified DDe u . . . ah.U reque.t tbroUCh

--

...

---."

If the report h •• been furnl.hed to the Office of Technlcal Service., Department of Commerce, for aale to the public, indl-cate thl. fact .nd enter th. price, if known.

IL SUPPLEIIENTARY NOTEI: U.e for .dclltlon,.1 .xpl.n .. tory note ..

12. SPONSORING MILITARY ACTlVlTY: Enter tbe name of the departmental project office or laboratory aponaoriDi (p.,..

I", lor) the r •••• rch end deve1opment. Include acldre ... 13. ABSTRACT: Enter.n .batract PV101 • brief end f.ctu.1 .ummatY ol tbe document 1odlc.Uve of the report, even thoulh

it may .180 .ppe.r .Ic.where 10 tbe body ol the t.c:hDlc.1

re-port. If .dclltlonal .pac. la raqulrad, • COftUnuaUon .he.t .haU be .ttacbed.

It i . hiply de.irable th.t tbe .bstract of cl ••• ifi*d reporta

be uncl •• ail1ed; Eech peraiNph of tb • • bstract ab.U .Ild wUb .n indlc.UOft of tbe mllitatY •• curitJ cl ••• UAc.Uon of t'"

iD-form.Uon In tbe peralf8ph, .rapre.nted •• ('1"). ('). (C). or (U). Tb.re i. no IimitaUon on tb.

leftlth

of the .bstract. How-.ver, tbe .ua •• t.d lenith la from 150 tI) 225 word ••

1 ... KEY WORDS: &e, word. &Ie tecbDlc.ll, mean1naful terma or .hort phra ••• that charecter1a • • report .nd m., be uaad •• index entrie. for c.talol1o, tbe report. Kay worda mu.t bs •• Iected .u tb.t DO .ecurity cl ••• iBc.tlOft 1. required. Idellt!-fi.re, .uch •• equlpment mod.1 de.lpaUon, trade nam., military proJ.ct cod. name, POiNPhlc 10c.Uon, may be uaad •• k., word. but wUl be foUo.ed by an 1ncIlc.Uon ol t.chDlc.l

con-text. Tb •••• ipmellt of UnIc., rul •• , .nd •• lpt. i. opUon.l.

UNCLASSIFIED

Security Cl ...

ification

l1rIAS Technical Note No. 146

Institute for Aerospace Studies, University of T oronto

~

Eddy Mach Wave N01se from a S1mpl1f1ed Model of a Supersonic !41x1ng Layer

Ribner, H. S. 6 pages 3 figures

1. Jet Noise 2. Rocket Noise 3. Acoustics 4. Aeroacoustic.

I. .Ribner, H. S. Il. l1rIAS Technical Not. No. 146

A simplified flow model is presented" tor simulating features of noise generation by supersonic jets a..'"ld rackets. 'Eddy Mach waves I appear as a consequence of b~anc1ng 1nternal a.nd external

pressures) and noise power may be estimated. Specifically, the turbulent mixing layer of' a

supersonic jet is modeled as a layer of two-dimensional square 'eddies'; this separates the

r.:ai" flow Uj from fluid at rest and moves at an intermediate speed. (In an improved model the

squa:e eddies are replaeed by a turbulent flow 'layer constrained by plain interfaces). The

ini tially plane interfaces, becallse of unopposed interna.l pressures, will ripple slightly such

t!:at Y~ch waves arise and effect a pressure balanee . The Mach wave noise pattern is readily

ealeulated as weil as the acoustic energy flux. In an example simulating e. round 'rocket jet

the effeetive mixing layer area i. taken to be 6 7T(diameters)2, the r.m .•• eddy·velocity i.

0.1 U.i' and Uj = 6 times external sOWld speed. The efficiency (noioe power/now power) come. out to 'oe &bout 1/310, which ls of the order of mea.sured values tor rocket Doise.

Ava'jlal:ile copies of this report are limited. Return this card to UTIAS, if you reqUire a copy.

l1rIAS TEX:HNICAL NOTE NO. 146

Institutefor Aerospace Studies, University of T oronto

Eddy Mach Wave Noise from a Simplif1ed Model of a Super sonic )lixing Layer

~

Ribner.) H. S. 6 pages 3 figures

1. Jet Noise 2. Rocket Noise 3. Acoustics 4. Aeroacoustics

I. .Ribner, H. S. II. l1rIAS Technical Note No. 146

A simplified flow model is presenteà.· for simulating features of noise generation by supersonic

jets a.'"ld rackets. I Eddy Mach waves I appear as a consequence of b~ancing internal and external

pre~sures, and noise power may be estimated. Specifically, the turbulent mixing le.yer of a

supersonic jet is modeled as a layer of two-diIDensional square leddies'; thls separates the

I:lain flow Uj from fluid at rest and moves at an intermediate speed. (In an improved model ~pe

sq".l.a!"e eddies are replaced by a turbulent flow layer constrained by plain interfaces). The

ini tially plane interfaces, beca~e of unopposed internal pressures, will ripple slightly such

that 1-1ach waves arlse ani effect a pressure balance. The Mach wave noise pattern is readily

calculated as weU as the acoustic energy flux. In an example simulating a. round 'rocket Jet

the effective mixing layer area is taken to be 6 7T(diameters)2, the r.m ••• eddy·velocity i.

0.1 Uv and Uj = 6 times external sOWld speed. The efficiency (noise power/flOW power) comes

out tI:> be about 1/3'{., which is of the order of measured value. for rocket noise.

l1rIAS Technical Note No. 146

Institute for Aerospace Studies, University of T oronto

~

Eddy Mach Wave Noise from a Simpl1fied Model of a Supersonic Mixing Layer

Ribner, H. S. 6 pages 3 figures

1. Jet Noise 2. Rocket Noise 3. Acoustics 4. Aeroacoustics

I. Ribner, H. S. II. l1rIAS Technical Note No. 146

A simplified flow lOOdel is presenteà' for simulating features of noise generatian by supersanic

jets and rackets. 'Eddy Mach waves' appear as a consequence of b~lancing internal and. external

pressures, and noise power may be estimated. Specifically, the turbulent mixing layer of a supersonic jet is modeled as a layer of two-dimensional square 'eddies'; this separates the

main flow Uj from fluid at rest and moves at an intermediate speed. (In an impraved model the

square eddies are replaeed by e. turbulent flow layer constrained by plain interface.). The

ini tially plane interfaces, beca~e of unopposed intern~l pressures, will ripple slight1y such

that Mach waves arise and effect a pressure balance. The Mach wave noise pattern is readily calculated as weU as the acoustic energy flux. In an example simulat1ng a. round ':racket Jet

the effective mixing layer area is taken to be 6 7T(diameters)2, the r.m ••• eddy·velocity i.

~.l Ui' and Uj = 6 times external sOWld speed. The efficiency (noise power/flOW power) come.

out tb be about 1/3'{., which is of the order of measured value. for rocket noi.e.

Available copies of

th

is

report: are limited. Return this card to UTIAS, if you require a copy.

JrIAS TEX:HNICAL NOTE NO. 146

Institute for Aerospace Studies, University of T oronto

~

Eddy Mach Wave Noise from a Simplified Model of a Supersoillc !41xing Layer

Ribner, H. S. _{6 pages} 3 figures

1. Jet Noise 2. Rocket Noise 3. Acoustics 4. Aeroacoustics

I. .Ribner, H. S. II. l1rIAS Technical Note No. 146

A simp1ified flow model is presented' for simulating :features of noise generation by supersarrlc

jets a."ld rockets . I Eddy Mac~ }rIaves' appear as a consequence of b~ancing internal and external

pressures, and noise power may be estimated. Specific8.lly, the turbulent mixing layer of a supersonic jet is modeled as a layer af two-dimensional square 'eddies I; this separates the min flow Uj from fluid at rest and moves at an intermediate speed. (In 8.n lmproved model the square eddies are replaced by 8. turbulent flow layer constrained by plain interfaces). The

ini tially plane interfaces, beca'.;1Se of unopposed inter~l pressures, nU ripple slightly such

that Mach waves arise ani effect a pressure balanee . The Mach wave noise pattern is readily calculated as weU as the acoustic energy flux. In an example simulating a raund 'racket ~et the effective mixing layer area is taken to be 6 7T(diameters)2, the r.m .•• eddy·velocity i .

0.1 U._{i ,} and Uj = 6 times external sOWld .peed. The efficiency (noise power/now power) comes