Refraction of sound by a jet: a numerical study

December, 1969.

REFRACTION OF SOUND BY A JET: A NUMERICAL STUW"

by

L. K.Schubert

Bibliotheek TU Delft

Faculte" der luchtvaart- en Ruimtevaarlteclviek Kluyverweg 1

2629 HS Delft

UTIAS Report

NO.144

AFOSR 69-3039TR

REFRACTION OF SOUND BY A JET: A NUMERICAL STUDY

by

L. K. Schubert

Manuscript received October, 1969 .

. ~

•

c.2

p. 8

E..:...2

p. 12 p. 18 ~ p. 30 p. 34 FIG. 7(b) FIG. 19 ERRATA u

2 shou1d be Uz in 1ine af ter Eqn. (1.2) a + D . shou1d be (a + U) . in Eqn. (2.5)

aXlS axlS

2

"2

a shou1d be a at the end of the 2nd paragraph

U. ~ 5.

2 shou1d be U. ~ U5.1 in 3rd 1ine af ter Eqn. (3.13)

l l l l

"Consequent1y (J .19) becomes " shou1d be "Consequent1y (3.29) becomes" in the midd1e of the page

[ltJ shou1d be[13J just before Eqn. (3.66)

submatrics shou1d be submatrices in 1ines No.6 and 13.

-1 [ 1

l

_{shou1d be} -1

[

:~

1

l

cos 1 + M. cos 1 + M.

J J J

ACKI'J"OWLEDG EMENT

The author would like to eÀ~ress his gratitude to Professor Ho S. Ribner, who suggested and supervised tbe resee .. rch. In the course of the research and in his critical reading of tb mo.nuscript he provi ed many valuable suggestions and insights that helped to sh.:lpe the contemts of this report.

Tha.nlu, are also p.ue to N. D. Ellis, L. T. Filotas, C. S. Matthias and Professor

B.

Etkin for their willingncss to discuss various aspects of the work and for their helpt~ul commentJ

enu.

ideas.

The resee.rph was sponLjorecl by t:1e N:ition':.l Resea.rch COWlcil of Canada.

under NRC grant No. A2003, and by the Air Force Office of Scientific Research,

Office of Aerospace Research, United states Air Force, pnder AFOSR grant

No

.

'.J

SUMMAHY

The equations appropriate to the propagation o~ sound in a realistic jet flow have been solved numerically i'or the C8.be or' a sinusoidal point source

on the axis of a subsonic jet. The computer program solves a transformed version of the convected wave equation by finite-di~ference methods~ yielding detailed phase and amplitude data throughout the sOUlld fi ld. Whe!"e comparison with ex -periment is possible ~ agreemerlt is genera.lly good~ although the numerical results

show somewhat more refraction. Confidence in the fini"te-d.ifference results is strengthened by their agreement wi th ray-tracing resuUis a.t the high frequency limit 0 The similarity o~ the computed directional curves suppor-iis the view that the downstream "va lley " in jet noise is due to refractiol'l rather than to the

inherent directivity of the sound geuerated within the region of turbulence. Sillce turbulent. scattering effects were suppressed in the numerical solution, the agreeme~t wi~h experime~t confir~s the expectation that these effects must be

relatively unimportant. Unexpected findings are that the distortion of the constant phase surfaces is slight

aqd

that the flow beyond lOO nozzle diameters continues to deepen the refraction valley sig~ificantly.

It has been necessary to dwell on the problem of formulating avalid corrvected wave equation at considerable length. In compa.!"ison with the wave

\

equation governing the acoustic pressure in uniform flow ~ the equa'~ion for non -uniform flow contains two extra terms involving gradients of the mean velocity

and mean density. The velocity gradient term~ which io. important at low frequencies, introduces an additional unknown into t.he equa·!iion., in th form of an acoustic

velocity component o &.'1. approximation to this term is possib1e~ but it doubles

the computing time and becCJ~s grossly in accurate for .Mach mmbers greater than

001. Because of these difficulties the wave equa.tion was reformulated in terms of Obukhov_{' s} "quasi-poterttial". This is easier to hand.le because no addi tional unknowns are introducedo Althoughthe equation is strictly valid only f'or uni

-form entropy and low Mach n~~er~ it yields results consistent with the pressure

formulati-on s:G high subsonic Ma(Ç;!h nUlllbers (at frequencies high enough to perrJlit omission of ·ehe gradient terros) and for hea'ced or cooled jets. It thus provides

a useful model for sound propagation in a. jet 0 Bath 'Versions of the wave equa-t ions, however , remain open to question whel1 the f'requency is 1011.' and the Mach number high ((vjDa

<

13 M

>

ol, where D is the nozzle di.a.r.1eter and ~. the so.und speed), although results in approxirr..a;i:;e agree:r.ent WHih experiment have been

com-puted in this regim.e. Also both versioll1> 3 ·l:ihough mutually consistent, exaggerate refraction for heated or cooled jets issuing horizontally. This is tentatively

attributed to buoyant eff'ects for real hea.ted or cooled. jets which spoi l the syrilme~ry by cUJ'ving the cen',,!"eline.

For the numerical solution~ q)..la!.1'Gities relatedèo 'che phase and ampli -tude of the acoustic distucQance were chosen as . dependent ïrariableso Unlike ;the a.coustic pressure ~ for example ~ these quantities are spa."iJially m.onotonic over regions large compared wi th the wa.velength. Thic fact facilitated the appli -cation of i'inite-difference methods~ even thoug~ the transfor:med wave equation is nonlinearo The difference equations approximat1ng the differential equation were sol:ved by nonlinear block relaxation usil'lg a Newton-like method. Because of computirig time considerations, accuraJGe solutions were calculated only within the f'requer;cy - lI..a.ch nuniber dom..q,in l' :< M:S .135

a

i

D.

lJess accurate solutions were

1. 11. I I l . IV. V.

VI.

TABLE OF CQNTENTS NOTATION INTRODUCTION 1.1 Background 1.2 Objectives 1.3 Method in Brief RAY TRACING

DERIVATION AND TRANSFORMATION OF CONVECTED WAVE EQUATION 3.1 Simplified Deriyation

3.2 Pressure Formulation and O~ukhov's Quasi-Potential 3.3 Shear Term in Pressure Equation

3.4 Non-Dimensionalization and Transformation to Polar COoordinates

3.5 Finite-Difference Grid

3.6 Grid-Optimizing Transformations 3.7 Finite-Difference Approximation

3.8 Difference Treatment of Bensity Gradient and Shear Terms ,

3·9 Boundary Conditions (1) 3.10 Conversion fvvm TI to p

SOLUTION OF NONLINEAR DIFFERENCE EQUATIONS 4.1 Methods Considered

4.2 Line Iteration by Broyden's Method 4.3 Block Iteration

RESULTS AND DISCUSSION 5.1 Ray Acoustics 5.2 v/ave Acoustics 5.3 Directivity 5.4 Phase 5.5 Summary of Data CONCLUDING REMARKS

APPENDIX A: Details of Ray-Tracing Formulae I'~. Normal Derivative of Wave Front

A2. Energy Flm-T Across Ini tial Waye Front A3. Evaluation of Sound Pressure tevel in Ray

rrracing

A4. Differential Equations for Single Ray Tracing A5. Stratified Medium

1 1 1 2

4

8

8

9

15 17 18 20 23 25 29 33 33 33 35 37 40 41 41 41 41 46 48

APPENDIX B: Integral Estimates of Sound Pressure Level at Small Mach Numbers

APPENDIX C: Moretti a~d Slutsky's Velocity Poten~ials Near a Point Source in Uniform and Jet-Like Flow APPENDIX D: Power Qutput of a Low Frequency Source in a Jet APPENDIX E: Velocity and 'l'emperature Profilèli'

APPENDIX F: Application of Refraction Results to Jet Noise

/ (

a, a , a. 0 _J

2:

a A A(P) 5A B B(P) C C. (i = 1,2, ... ) l. d(P) D D/Dt 2), ~ .. l.J e E f f( ),f'( ), f"( ) f(P) f (i,j) re f. _l.m (i,j) F H i N0rATION (Excluding App. C)

local, ambient, and jet exit sound speed local time-averaged square of sound speed amplitude of p(l)gr

TI

=

IBI

th . t' t J b' P approxl.ma l.on 0 aco l.an

element of area

( 1) iwt iwt i <IJ compiex amplitude

=

p e or

TI

e

=

Ae pth approximation to inverse Jacobian J-l a./a

J 0

constants or parameters a correction to x(p)

jet nozzle diameter

convected time derivative determinants

power between two ray cones

ei

frequency in hz various functions

d/dt

+ v.

d/dX,

l. l.

vector function consisting of the f , f. on

t

grid 1 , l.nes, w h en th e correspon lng un owns are approxl.ma e d' kn re l.m . t d by x(p)

point residual associated with real part of difference equation at grid point (i,j.)

, /

point residual associated with imaginary part of difference equation at grid point (i

,D

.

)

2 2

[R /(2K)] ,

a parameter in the difference equations a finite increment in ~

I k L m M

M

.

J n p, p ( 0) p

pel)

P, Q,

R

q r R.(i ;:: 1,2'0000) ~ s /,),/,) 0 S t

t

t(p) T u. (i ~ 1,2,3)

energy flux density

Jaeobian of the funetions _{f re , f im} with respect to the grid point values 84 " 'f' . . (i ;:: 1,2,000.,m ,j ;:: 1,20.,n)

... ,J ~,J wave number =

wja

finite increments in

e

or

S

the number o:f grid lines in a relaxation bloek

len~th of simplified jet flow model number of circumferential grid lines loeal Mach number

jet exit Mach number

~umber of radial grid lines

11 a function of the order of .. . o • • "

loeal instantaneous statie pressure, loeal mean pressure

statie pressure without injeeted sound

aeoustic pressure perturbation due to injeeted sound

constants or parameters

amplitude of volume flow from a souree

dimensional radial distance from souree point parameters in the difference equations

1) entropy (in the partial differential equations); 2) distance along a ray (in ray tracing context)

source strength;:: amplitude of rate of change of mass

flow from a source

~ i~

(x/x

)

=

relative sound pressure level (in the

pressure-~ormulation)

time

t -

rja

a scalar used in quasi-Newton methods

2 2 2 2 .

(a -a )/ (a. -a ) = normal~zed temperature ratio

o J 0

acoustic velocity perturbation due to injected sound

U. (i 1.

1,2,3)

U. J v. (i _1.

=

1,2,3)

V V f V s w. (i _1.

=

1,2,3)

w,

W , W. 0 _J x

-)p) X, Y I z

ex

t3

)' 5 .. 1.J E

S

Tl 9 À

ç.

(i 1.

1,2,3)

Çz TI -p,p p (0 )

mean velocity (local time average) jet exit velocity

instantaneous velocity

mean velocity ratio in a jet phase velocity

ray velocity

v. - u. _{1. 1.}

wD/a, wD/a , wD/a. (wD/a -

kD

=

2nD

/

À)

o J 0

location of field point

a column vector approximating the sol ut ion to the difference equations on $ grid lines

J

'

2 2

l-M S1.n 9, Mcos9 (see Fqn. 3.64)

location of source point

J

x 2

2 _{+ x}

3

2

_{= lateral distance from jet axis}

a consta.I7t (see 'Eqn. 3.36)

a constant (see Eqn.

3

.41)

ray angle relative to jet axis Kronecker delta

energy density

tan-l

(ex-

2 tan9)

angle of radial line with jet axis wavelength

x./D nondimensional Cartesian 2coordinates

1.

ç

2

3

quasi velocity potential local, loc al mean density

(J cp

x:

,

X

0

w

density perturbation due to injected sound r/D

=

nondimensional radial distance

direction of wave normal relative to jet axis phase of p(l) or TI

=

arg

B

(JA ; a reference value for X

radian frequency of sound

angle with the vertical in a plane perpendicular to the jet axis

---

-

--

---proportional to

approximately proportional to

I. INTRODUCTION lol Background

In

1963

a series of experiments were begun in the anechoic chamber of the Institute for Aerospace Studies to investigate the far-field directivity pattern of a "point" source of sound imrnersed in a subsonic jet flow [1-5]. The observed pat-terns showed the same pronounced axial intensity minimum that is characteristic of aerodynamic j~t noise. This similarity lent strong support to the view

[6-8]

that the downstream "valley" in jet noise is due to refraction rather than to the inherent directivity of the sound generated within the region of turbulence.

The present study is an analytical counterpart and extension of the

experimental refraction results. The study was suggested by Dr. H. S. Ribner, with motivation by the following objectives.

1.2 Objectives

It was hoped that an analytical model incorporating the most important features of the physical system - particularly arealistic velocity field - would overcome the following limitations of the experimental data:

(i) No phase information for the acoustic field was available. Without this in-formation it remained uncertain how the refraction results for a stationary,

harmonic point source should be applied to the prediction of jet noise directivity, since jet noise is generated by distributed, moving sources. Previously the flow had in fact been assumed to have zero effect on the phase.

(ii) The experiments gave only the amplitude of the pressure fluctuations 100 jet nozzle diameters from the source. More complete amplitude information showing the development of the refractive mini~ was lacking. Also, the spatial limita-tions imposed by the anechoic chamber made it experimentally difficult to validate the assumption that far-field conditions prevailed 100 diameters from the source. Quite possibly the axial intensity could be appreciably depleted by the slow but spatially extensive flow further downstream.

(iii) No data for a frequency - Mach number product greater than .15 aiD were available* because of the very low signal-to-noise ratio for the injected sound field near the jet axis under those conditions.

(iv) Because of the limited power supply available for heating the jet, very hot jets could be obtained at very low Mach numbers only.

(v) Some doubt remained whether turbulent scattering effects were indeed negli-gible in comparison with mean floweffects.

In addition to extending the data for refraction by a jet, a sufficiently general techntque for computing the sound field of a point source in a steady flow could be used for exploring other flow configurations, e.g. a flow advancing to-wards the source so as to produce acoustic focusi~.

Apart from practical implications, the problem of wave propagation in a * this way of expressing a bound on the frequency - Mach number product mayalso

be written as M(fD/a)= M(D/À )

=

constant; the ratio D/À is a yardstick for the refractive effectiveness of the flow.

medium with velocity and sound speed gradients th at are non-uniform in direction as well as magnitude is interesting in itself since only cases with unidirectional gradients appear to have been treated so faro

1.3 Method in Brief

+he analytical model consists of a harmonic point source located on the axis of an axisymmetric flow field (Fig.l). The functions chosen to simulate the axial and transverse mean velocity profiles closely approximate the mean velocity profiles of a real jet

(App.

E; Fig.2); it is assumed th at the radial velocity components are small. TIhe temperature profiles for a hot flow are based on the proportionalities

[8]

T ~ ~ transverse to the jet axis and T ~

V

along the jet axis, where T

=

(a2-a 2)/(a. 2 - a 2) and V

=

velocity ratio U/U,. The physical model in principle a~ts turbul~nce, but turbulent scatteringJeffects are suppressed in the final wave equation. No solid reflecting surfaces are permitted in the model. For the sake of flow continuity an unconfined convergent flow which supplies the jet flow may be presumed to lie upstream of the nozzle plane. The exact nature of this "antijet" has been found to be unimportant as far as the downstream sound field is concerned. A momentum antijet and a constant-mass-flow antijet gave virtually indistinguishable results.

For the limiting case of very high frequencies one can employ ray-tracing techniques to compute the pressure field of the source. These techniques turn out to be quite adequate when the diameter of the jet is several times the wavelength of the radia~ed sound. Such geometric acoustics solutions were programmed for comparison with the wave acoustics sol~tion: they should approach agreement at the high frequency limit.

At lower frequencies diffraction and reflection effects are prominent,

~d it becomes necessary to solve the convected wave equation. The simplest form of this equation is[e.g.

7,

11-13,16]

+ 2U - a

2'

-2 V p (1)

o

( lol)

where the flow is in the Xl - direction at speed U. This is strictly valid only in a region of uniform flow but it has also been applied to nonuniform flows U

=

U(x_l'x

2). Several solutions of this equation for simple flow fields are avail-able, dea11ng with configurations such as a pl~ wave impinging on a planar velo-city discontinuity [11], a line source near a planar velovelo-city discontinuity

[12,13] and a point source near a cylindrical velocity or temperature discontinuity [13 to 15]. In all cases the velocity discontinuity extends to infinity, so that the refractive effect is exaggerated by orders of magnitude in comparison with a real, finite jet.

A real jet flow possesses three contiguous flow regimes (mixing region, transition region, f~lly developed jet) each of which is characterized by its own

complicated dependence on radial and axial c~~rdinates

(App.E).

This complicates the convected wave equation in two respects.

First, the coefficients are no longer constant, but functions of two variables. The analytical techniques employed in the idealized problems cited above all rely on the constancy of the coefficients except at discontinuities

..

"

that are independent of two cooordinate directions. A closed form solution in terms of realistic jet velocity functions therefore seemed to be an impractical goal; a computer-orientèd finite-difference formulation was considered to be the most promising approach.

The second complication of the wave equation is the appearance of new terms proportional to the gradients of the mean velocity and density. Specifi-cally,it is shown in Part III that the convected wave equation for the pressure becomes approximately 1

2"

a. + 2U ~

ou

~p-- oU z 1 0 2p

dz

oX l - ij dXi ( 1.2) where z

=

J

x 2 2 + x

3

2 and u

2 is the radial component of the acoustic velocity. ~he main assumptions leading to this equation are that the effects of viscosity,

heat conduction and turbulent scattering can be ignored (in effect, these effects are suppressed by the construction of the equation), that the fractional pressure variations in a jet are small compared to unity, and that axial velocity gradients

are small compared to radial velocity gradients. The density gradient term on the RHS of

(1.2)

presents no special difficulties and is in fact found to become important only at frequencies below those relevant to jet noise. The velocity gradient term, however, is troublesome because it introduces an additional unknown, viz. u , into the wave equation. At low Mach numbers it can be taken into approxi-mate a~count in two different ways. In the context of the integral approach dis-cussed in Sec.

3.3,

it can be treated as a source term; this method gives some indication of its effect, but is quantitatively unreliable. In the context of the

difference approach th at is the main subject of this investigation, it can be

approximated by recourse to the momentum equation (~.3.8). The use of these approximations has shown that the shear term cannot be ignored for wavelengths

À

>

2D. lts inclusion in the difference approach doubles the computation time,

and in any case the approximation breaks down for

M.

> .1.

J

terms of

These difficulties prompted a reformulation Obukhov' s "quasi-potential" IT defined by

u. l

t

dIT

J

( dW i dIl dW j dIT )

dX."

+

dx."

dX:" - dx.-

~

l J J l J

of the wa~e equation in

dt (1.3)

where w. is the local instantaneous velocity in the a~sence of injected sound. With

th~

assumptions that the entropy is uniform and M is small, the wave equa-tion for IT turns out to ~e

i

w (1.4 )

The RHS of this equatiop, unlike that in

(1.2),

does not introduce an additional dependent variable, and its importance diminishes more rapidly as the frequency is increased. The wavelength at which it begins to play a significant role lies in the vicinity of À =]D. For these reasons the IT-formulation is more conveni~nt

found to be consistent with results based on (1.2), the wave equation for the pres~ure, for all cases where comparison is possible. Surprisingly~ this includes results at Mach numbers that are not small (at frequencies sufficiently high so that the terms on the RHS of (1.2) can be neglected), as well as results for heated or cooled jets, for which the e~tropy is nonuniform.

It thus appears that the quasi-potential formulation as expressed in

(1.4) is an adequate mathema"tical "model" for sound propagation in a jet. Some doubt remains whether this is true at low frequencies (À

>

70) and high subsonic

Mach numbers. Judging from experimental res~lts, the best results in this regime appear to be obtained by using a zero R5S in (1.4). Also, results for heated and cooled jets obtained with the pressure or quasi-potential formulation, though consistent with each other, show much stroneer refraction effects than are

oQ-served experimentally. The discrepancy is tentatively attributed to the convective upswing or droop (for heated and cooled horizontal jets respectively) which destroys the axisymmetry of the flow agd may thus reduce refractive effects.

rh~ Qumerical method used to solve the wave equation was the same for both the p\l) and TI - formulation, with and without gradient terms. First the time dependence e- iwt was factored out. The depe~dent variable in the resulting linear complex elliptic partial differential equation is B (the complex amplitude of p(l) or TI). The use of the real and imaginary parts of B as dependent variables is poorly suited to the application of finite-difference methods: they would require an inordinate number of grid points to delineate because of their sinusoidal charac-ter.

To avoid this basic difficulty in the linear approach B was replaced by another pair of dependent variables , one ampli tude-related, the other phase-related, and both being monotonic over regions large compared with the wavelength. This made it possible to use apolar finite-difference grid (lying between a very small and a very large source-centered spherical surface) with grid spaces increasing in direct proportion to radial distance. The price paid for the transformation of B is the nonlineari ty of the resuLting pair of equations. F'urther transformat ions, which compress the grid near the axis to resolve rapid variations in the acoustic field there and express the amplitude in decibels, did not alter the character of the equations. The transformations are summarized in Fig.

3.

The system of nonlinear difference equations (578 equations in most cases) approximating the differential equations were solved by nonlinear block relaxation using a Newton-like method. The number of unkno~s in a block ranged from 34 (1 radial grid line) to 238 (7 radial grid lines).

The development of an acceptable wave equation and method of solution has involved a "feed-back loop" wherein the domains of validity of various assump-tions and approximaassump-tions could be ascertained only by inspection of actual computed results. Consequently it has ofte~ been necessary in the sequel (particularly Part 111) to cite such results prior to a proper discussion of how they were ob-tained.

11. RAY TRACING

Derivations sketched in this section are outlined in more detail in App. A.

When changes in the properties of the medium are negligible over a

that is the vector sum of the local sound speed directed along the wave normal and the local velocity of the medium [16, 17]. Hence for sufficiently high frequencies one may readily trace the propagatio~ of a wavefront from the vicinity of the source into the arnbient medium.

The known solution of the convected wave equation for ~iform flow [e.g. 13] was used to calculate the energy densities on the initial wavefront.

For computational purposes the wavefront can be identified with the end points of an ensemble of rays, which are jointly incremented in small time steps.

----...

Sketch 1: IncrementatioB: of rays in "batch ray-tracing"

If ~1'~2"" " "~5 are the distances of 5 neighbouring ray termini from the source at a particular time, and Kl,K2,K ,K4 the angular separations between them, the slope of the wave

~ormal

for the trtird (central) ray is approximately equal "tio

tan 8 3 - 1

where ~3 ' is the fourth order polynomial approximatiàn to d~/d8 in terms of the

~. and K. (details in Appe~dix Al). Accordingly each ray increment may be expressed ik termslof 5 local ray termini and the local flow velocity. Such a procedure was coded in FORTRAN IV for the IBM 7094 computer, utilizing up to 33 rays distributed from 00 to 1800 with the jet axis. Mirror-image rays were used to complete groups of

5

near the axis.

'; determine the conditions necessary f'or computing reliable sound pressure levels.

These levels were estimated from adjacent rays (designated by subscripts 1 and 2) using formula (A15) of Appendix A3. The far field version oLr this formula; at

low jet Mach numbers, reads

{ cosep{ - - cosep2-- } SPL ((J,8)

~

CONST. + 10 loglO

~2

u cos8 1- cos82 (2.1)

where (J,8 are source-centered ,J:>cordinates and cp I is the direction of the wave

normal at the ray origin; (J

=

((Jl + (J2)/2, 8

=

(8 + 8₂)/2. The above expression

is obtained by dividing the acoustic energy flow

~etween

two ray cones

(cos epl-- COSCP2-) by the normal area between them.

The technique of tracing a la.rge ~umber of rays simultaneously, though relatively simple and efficient, f'ails when rays intersect. This may occur for rays travelling upstream at small angles with the jet axis or travelling downstream in a very cold jet, such that they are refracted back towards the axis. To

overcome this difficulty, one may trace rays individually. Various formulations of the dif'f'erential equation of a ray can be used to this end.

Consider the small line-segment of a wave front lying between two

near-by rays in the x_l-x2 plane. The angular velocity of this segment is the difference in phase velocity between the ends of the segment divided by the length of the

segment.

j;,.'",

" <

'

~blJ

I'

>

.(a

dU

cos

ij>

I

d (differences due to the

L _ _ _ curvature of the wave

U

C05CP U front are second-order)

Sketch 2: Determination of' angular velocity of wave front

Referring to Sketch 2, this is -(5a + 5u cosep)/d or

v

s

-

[

dep = ds 5x

l = - d sinep, 5x2

=

d cosep we have

dep àa àa ( (jU . àu

dt =

dX

l sincp -

dX

2 cosep + coscp

dx

l sJ.nep -

öX

2 substituting cosep ) (2.2)

where s is measured along the ray and Vs is the ray veTocity. The same result can be obtained somewhat more rigorously from the generalized Snell's law (Appendix A4). The equation for dep/ds and the geometrical identities (where

M = Mach no.) sinep

v

s a

v

s a sin, , cosep

=

v

s - cos, - M a

=

J

1 + 2M C0Sep

+

M2

(

2

.

3

can be used to derive expressions for the derivative of the ray.angle with respect to s in terms of ep or , : =

=

2 a V2 s 1 V s 1 V s

r(l + Mcosep) dep _ sinep

t

ds

~J

2 [ a . èu 2

~

MSln

3

ep

~

+ (coscp

+

2Mcos

~M)

V oX_l s

{ :s sin, [ èa _è)x M . 2 èu

] +

[2Msin2, _ a l Sln , è)x l V S .2 [ - - 1 + M 112 _s ( : s cos, _ M

)3

] èu

dX

₂

}

n ] èa _. :B~ ":. cos, è)x 2 V_-;;; ~ (2.4) Since these expressions give the local ~urvature of a ray, they may be used for ray tracing. In incrementing a ray from a point A to a pmmnt B, one may adjust the local step size until the discrepancy between two estimates of the ray angle at B, namely ' 'B ~ 'A + 5s (d,/dS)A and

'B

~ tan

-1 { sin-[CPA + 5s

(dCP/dS~]

J}

.

~ + cos[CPA+ 5s(dCP/dS)A

~ecomes sufficiently small compared to 5s, the acoustical path length from A to

B.

To obtain approximate far field intensities one may proceed as follows. Initially space the rays such that xhe same energy flow is ass0ciated with each. Then trace each ray until it suffers no further deflection. If the final ray

angles of two ultimately neighbouring rays are

'1

and

'2'

then sufficiently far from the flow the intensity at

,=

'1

+

'2 wiIl be 'approximately proportional

to 2

1

Intensities in the region where rays in~ersect were not calculated because of the aWkwardness of the computation and the necessity for very large ray densities.

Some results of "batch ray-tracing" and "single ray tracing" are shown in Fig's.

4

to 6.

Apart from furnishing directivity patterns for very high frequencies~

ray acoustics also permits certain inferences regarding the decay of the axial sound field very far from the source at arbitrar~ frequencieso No matter how

large the wavelength, the cross-section of the flow eventually is much larger

and velocity gradients correspondingly smallo Thus ray acoustics becomes appli

-cable and the axial phase variation is asymptotically proportional to the ray

-acoustic phase variationo 'rhe phase retardation relative to quiescent conditions

(or relative to conditions at

e

= 900) at some 3ufficiently large distance Xl downstream from the source is therefore

X o dx l a+D . ax~s

.

I

-1 2 2 ( 2 2) -1

ass~ng th at Uaxis Uj ~ C2 Xl and aaxis- a_o ~ _C2 a_j -aa Xl ; note that

C

=

a./a ; C

= 6.39 when distance

s are expressed in nozzle diameters. It is

evide~t ~r

o

m

this formula that the axial phase differential increases forever.

However~ the ratio of the differential to Xl approaches zero~ so that constant

phase surfaces of increasing radius become more and more nearly spherical.

As far as the amplitude of the axial pressure is concerned~ it is found from ray-tracing results that the depth of the refraction valley approaches an

asymptotic limit. As an example the refraction patterns for a Mach ,05 jet 100 D from the source and "infinitely" far away are shown in Fig.

5.

The additional

refraction beyond 100 D amounts to over

5

dB. The difference becomes 9.5 dB and

18.2 dB for Mach .175 and Mach .3 respecti vely. Since these figures are probably

roughly correct for all but the lowest frequencies (e.g.~ assuming ray-acovstic

propagation beyond 100 D for À

<

10 D) it is by ~o means safe to equate refraction at 100 D to tha.t in the true far field.

111. DERIVKÇION AND TRANSFORMATION OF CONVECTEg WAVE EQUATION

3.1 Simplified Derivation

This sU9section is intended as an abbreviated introduction to the wave

equation in the form used for most of the numerical work, A more accurate equa

-tion is d rived subsequently.

Consider an unaccelerated frame of reference instantaneously following

a fluid particle in a flow. One might expect the quiescent wave equation (for a

source - free region)

to be satisfied in the vicinity of th at particle. Viewed from a stationary frame

of reference (3.1) takes the form

1

2'

where DjPt

=

o/dt + v. O/ox. is the time derivative following the flow. The term

a-2Dv.jDt x op(l)

jox~ resu~ting

from-hhe second

applicat~on

of the DjDt operator

may

b~

_{neglected, as rt is equal to -(a2p)-1(OpjoXi ) (dp(lJj OXi) by virtue of the}

momentum equation and is thus of second order. If in addition the flow is in the xl-direction and turbulent scattering and sound generating terms are suppressed, (3.2) becomes

1 _{(o2p (1) 02p (1)}

~ 2 + 2U

a 0 t oxlot

The chief flaw in this oversimplified deduction is that the effects of the mean velocity and density gradients have been ignored in (3.1), so that (3.3) is

strict-ly correct onstrict-ly for a uniform flow. In spite of this inaccuracy Eqn. (3.3), which

merely describes the downstream convection of the sound field when the flow is

uniform, acpounts for most of the refraction due to spatial variations of_U and

a2 in a jet flow.

Note that the entire argument could also be framed in terms of a velocity

potential TI. In actual fact, however, no velocity potential exists in a vortical

flow. Nonetheless, Eqn. (3.3), with p(l) replaced by TI, has been successfully

employed for shear-flow problems by other workers [18J.

3.2 Pressure Formulation and Obukhov's Quasi-Potential

First a fairly accurate formulation of the convected wave equation in

terms of pressure will be derived. It will then be shown that at low Mach number

and constant entropy a formulation in terms of a quasi-potential TI is possmole: .

This formulation appears to have wider validity than these restrictive assumptions

suggest.

In the wave equations terms involving non-axial and turbulent flow

velo-city components will be neglected.

It should be noted that the neglect of turbulence terms in the wave

equa-tion is not necessarily equivalent to the neglect of tur~ulent velocity

fluctua-tions in the physical model. The use of a time-averaged, unidirectional flow as

model would involve a violation of the fluid dynamical equations. In particular,

continuity would be violated since the flow decelerates in a stream tube of constant

cross-section; the momentum in such a stream tube would change without the action

of a pressure gradient; and for a hot time-averaged jet the substantive derivative

of the density would fail to v~ish.

Thus the model for the subsequent discussion fncludes turbulence, up to the point where turbulence terms may be safely discarded.

Viscosity and heat conduètion - and more especially, molecular

relaxa-tion effects - account primarily for the attenuatio~ of sound at large distances,

over and above the inverse square law. Furthermore, the roles of viscosity and

heat conduction in the transport of momentum and energy in the jet are very minor

compared to that of turbulent mixing. They will therefore be neglected herein.

Dv. ov. ov.

~ ~ ~

-

- + v .

-

-Dt ot J ox.

J

Differentiation with respect to x. yields

~ o

(dV

_~

.

)

₊

ov. J ~

dx."

dx:-1 ~ 1 1 P

~

.

~

dv

i

d

( l O P )

dx-:-

= - dj{.

P

dx. .

J ~ ~

Substitution of the continuity equation in the form

then gives D Dt

dV

~

.

di":"

=

-~ 1 P Dp Dt ( 1 Dp ) ov .

Cv

i - - + -P Dt

df

Ox. 1 J 1 Op

""'2

~ _ox. p ~

~-

~ 1 P

dx

_~

.

dx

_~

. .

(3.4)

The neglect of viscosity and heat conductio~ leads to the simple energy equation

[ 16]

Ds

Dt

=

0 (3.8)

Consequently the quatio~ of state₃ applied to a moving partiele of fluid3 becomes

Dp

Dt One Ca.l1 therefore expanJd the first

D ( 1 DP) 1

D~

Dt

P

Dt

=

pa2 I?t2 1 D2p

2

Dt2 pa 1 D2p

2

Dt2 pa 1 ~

""'2

Dt a. term in

(3

.

7)

as 1 (D)2 1 Da-2

24

\~

+

P

:::'"Dt:----p a Dp Dt

).4

(~r ~;~4 (~)2

ja4(~)2

(3.10 )

where the second line results from the isentr~pic relations for an ideal gas 5a-2

=

5(yRT)-1= - (yRT 2)-1 5T = - p (yRT2p C

v)-l 5p

=

-

p(YRT2p2

C

va2)-1 ~

5p = - (y_l)/(pa

4)

5p.

The second term of the RHS of (3.10) can be neglected if it is supposed

that fractional pressure changes in the jet are very small, i .e., ~p-p )/p =

E f(x₃t) where E

«

1 and f ~ 0(1) . In that case Dp/Dt ~ Ep wf, D p/D€2 ~oEP w2f

- 0 0

and the ratio of the second term to the first ~ (fp lp) E « 1 . This is rather a loose argument, si~ce the symbol p mumps together aePodynamic ~d acoustic components. In a more careful treatment one finds that the quadratic term in (3.10) merelyadds terms involving mean pressure gradients to the wave equation, and that all of these terms can be neglected under the somewhat more str~gent assumption that fractiona1 mean pressure variations are small compared to variations in the mean velocity ratio

*

Substituting (3.10) , without the quadratic term, in (3.7) one obtains 1 D2 d2

dv.

dV. 1 dP

~

~ P 1.

öf

(3.11)

2"

_a Dt2 - dX_i dX_i = P ~ ~ J 1. P 1. 1.

This of course is equivalent to Lighthill's equation [20] and i~s variants, a1though the demoll8tratio~ of equivalence in the gen

₂

ral case (nonuniform mean density)

by expansion of the Lighthill source term d (pV.V.jdX. dX. and conversion from density to pressure on the LHS is more difficult thaa1. t heJderivation of either equation. O. M. Phillips: variant of the wave equation [21], however, closely reseIDbles the present version. Note that the RHS of (3.11) includes not only sound generating terms, but also sound propagating terms, such as PdUjdx₂du

2jdxl (where U and u

2 are mean and acoustic velocity components). When DjDt is expanded (3.11) becomes

",2 ) o P + v. v. _{1. J} ""'"à'-"""à-_{x. x.} -1. J

d

2 _P

₌

_p dx.dx. 1. 1.

dv.

_1. ~ J (3.12) The term Dv.jDt x dp;dx. resulting from the second appli~ation of the DjDt opera-tor has beeft discarded, §ince it is equal to _p-l(dP/dX.) by (3.4) and is thus a second order quantity. lts retention would lead to further terms in the final wave equation involving mean pressure gradients. Like those resulting from the retention of the last term in (3.10), they turn out to be neg1igible.

One may replace

p,

v. and p by

p(o)

+ p(l), w. + u. and p(o) + p(l) respectively. The first part 6f each sum corresponds t6 the1.value in the absence of injected sound; the second part is the additional perturbatio~ due to the in-jected sound. Neglecting second order terms and products of acoustic perturbations with pressure gradients, and subtracting the equation valid when no injected sound is present (this automatical1y eliminates terms associated with the generation,

prppaga~ion and scattering of aerodynamic jet noise), one obtains

)-*

-tJ.p/po ~ Po *~É.. Townsend [19] gives a maximum

Vi

2_{/Uj 2} = .• 0056 at Sl = 20. This corresponds to a fractional pressure change .0056 ,(U.ja )2 on the axis.

êlw. ( ) êlw.

di

+ 2p 0 1. ~x. X. OX. 1. J êlu.

öt-

_1.

-::TöT

1 _p êlp(O) dX. 1.

Scattering Of(tbe injeeted sound by the turbulenee ean now_be suppressed byre-placing a2 , pO), and _wi by their local time averages

;;:2,

P, and U.o If in

addition Ui ~ 5_i2 (i.e. unidirectional flow) then (3.13) becomes 1.

-L.. (

ö

2p(1) + 2U êl2p(1) + U2 êl2p(1)

)-o2_p(1)

2"

êlt2 'dXldt oX l 2 dx.dX. 1. 1. a

(1) (êlU )2 _2p êlu êluJ . 1 êl

p

êlp(l ) _(3.14)

= p

di

+

_dX:"

_dX

l -

dx:-

d

xl _{J P 1.} x_1. .

Since the flow in a jet is nearly parallel, êlujdxl is small (/êl(u/uj)/êlsll< • 1

for the jet; by contrast lêl(u/u.)/à~2

I

~ 102 at the end of the mixin~ region)

so that it seems reasonable to ~etain oWî~ 2p êlu/ozou /êlx

l , where z =(x' + x 2,

of the seeond souree term in (3.14) . The first

so

ure

~

term(i~ appreciabîe(o~~y

at very low frequencies, as may be ascertained by writing p 1) as (~)-l p 1) and

comparing with the first term of the LHS (term is important for

w

~ êlu/êlxl ' Le 0' f < .016 Mja/D; e .go, at Mj

=

05,

IJ

= .75" the threshold is f

=

142 hz).

1

2'

a

Thus one arrives at the approximate wave equation

)

-

-p êlx.

1. êlx1. .

Together with the momentum equatio~ and the boundary conditions this equation

de-termines the acoustie pressure and velocity disturbance.

Strictly speaking, one cannot solve the governing equations in this form

for the pressure without solving for the acoustic velocity as wello The obstacle

is the shear term 2p oU/êlz x. êluz(êlx

l in (3015), which cannot be expressed in terms

of pressure alone (exeept a.s an integral) 0 It contains oztly a singly differeI}tiated

acoustic perturbation quantity, whereas all terms on the

L

H

S

of (3015) contain double derivatives. This suggests that the ratio of the shear term to any term

on the LHS goes roughly like w- l , i.e., it becomes increasingly important at low

freque~cies. This surmise is borne out by the results cited in the more detailed discussion of the next sectiono Similarly the densi ty gradient term is of order

w-1 . Consequent1y the two gradient terms caQ be omitted from (3.15) for sufficiently

high frequencies. It turns out, however, that the shear term is still significant at frequencies high enough to be of interest in jet noise research.

in terms of some quasi velocity potential. The resulp is a form of Obukhov's equation, as will be shown.

Substitution of v. = w. momentum equation

(3.4)

yi~lds l

ou. l + u.

dx."

J J ow. l

dX:"

J + u., P l (3.16)

af~er subtraction of the corresponding equation fof the undisturbed flow. Simi-larly the continuity equation (3.6) becomes

Op(l)

2ît

+ w. l ou. l + dX," l ( 1) P Ow. l ~ l

o

(3.17)

Now it is necessary to assume constant entropy. In that case, the sound is also isentropic and (3.16) and (3.17) become

ou. ou. ow.

0

(~)

l + w. l + u. l

d t

J

dX:"

J.

ë1X:"

~ D- J l 1 (op (1) dP(l) ) (0) ou. l= 0

2

dt + _wi ~ +p dX," = a l l

Terms involving

V .

~

and derivatives of p(o) have been neglected in (3.19) as these quantities are small in an isentropic flow of nearly uniform pressure.

We wish to express u. and p(l) in terms of some scalarrTI and thus obtain

t~r)differe~tial

equation of ln. A natural choice for the relationship between

p and n lS

(1)

N

p

(3.20)

which is the equation connecting pressure and potential in an irrotational flow. This ensures that n can be identified with the velocity potential wherever the

latter exists. No velocity potential exists in a rotational region of the flow; in general the true acoustic velocity therefore differs by some amount u~ from

-VIT

i.e., l Substituting Ou.' . l ' + w. ~ J u.

=

-l on ,

dx.'"

+ l1_i l

for p(l) / p(o) and u. in (3.18) ,

l

ou' ow. ow. on ' i '

,

l l ~ + u.

dX:"

'dX:" 'dX:"

-J J J J J (3.21) :''.;

we obtain an equation for u' . l ow. _on

ö1

(3.22)

Ox. l _J

Unfortunately, this cannbt in general be solved for u. I . A first approocimation

when wand its derivatives are small is obtained by n~glecting the second and third terms, which leads to

u ' i t

JC

dW.

d

IT

~

.

dX~

dX, J J

dW

j

d

IT

)

- dx

.

dX."

1 J dt

This is Ob~nov's approximation [16]. That it is a valid first approximation is

readily verified by substituting (3.23) in the neglected terms of (3.22) to

ob-tain an improved clpproximation to u.' and noting

th~t

)t

he

new terms are

quad-ratic in

~

and its derivativeso SuEstitution for

p~l

and ui in (3.19) gives

Obukhov' s equation (withthe density nearly uniform):

t~2

d

2

n

)

d

2

IT

J

ow. dXidX_j - dXidX i - dxigxi

dIT

dt == 0

dX:"

J (3.24)

A term quadratic in w has been ignored. When turbulent scattering is neglected

and the mean flow is-in the xl-direction this becomes

1

C

d

2

IT

d

2

IT

2

d

2n2"

d

2

n

i

d

2U

~ ~t2

+ 2U dXldt + U

J

dx.dx. w dx.dx. 2 u dX l 1 1 1 1 a

d

IT

dX l

=

0

assuming a sinusoidal sound source. For constant density this equation is

identi-cal in form to the pressure equation (3.15), except for the extra term involving

the Laplacian of the mean velocity.

The significant feature of the present equation is that the extra term is of o~der

w-

2 in comparison with the other terms, whereas the shear term in

the pressure equation is of order w-lo . Therefore as the frequency is increased, the contribution of the extra term to

(

3

0

25)

falls off more rapidly than that of

the shear term in (3015)0 This might lead one to think that the two formulations are contradictory, since the differential equations differ only in the extra termso The apparent paradox is resolved when one considers that different boundary

con-ditions at the source apply t o the two formulations (see Seco

3

0

9)

0

In other words~ at some frequency where the extra term in the potential formulation is n

eg-ligible whereas the shear term i~ the pressure formulation is non-negligible, the

difference in boundary conditions for the two cases exactly cancels the effect

of the shear term in the pressure equation*o

This conclusion is amply supported by numerical resultso An illustration is given in Fig080 for the case Wo == 1.055, M. == .025. The contributions of the

extra terms are seen to be very different forJthe two formulations ,yet the final

curyes are quite similar.

For cases where it would be necessary to include the shear term in the pressure formulation, the potential formulation provides an attractive

alterna-tiveo The advantage is that the extra term can then be omitted down to lower

frequencies and that its incl~sion, if required, is a simpler tasko"

In fact, the potential formulation without extra term has been found to be consistent with the pressure formulation and with experiment even for high subsonic Mach ~umbers (at least for unheated jets) o Most of the data in the present report have been computed on this basiso The use of the potential formu-lation at high Mach numbers has been validated by comparison with results for the pressure formulation at frequencies sufficiently high to permit omission of

*

Mo~e)accuratelY, exact cancellation occurs only outside the jet flow, where

the extra terms from both formulations. To determine the quantitative meaning of "sufficiently high" a considerable effort has been applied to a. study of the shear term in the pressure formulation. One line of attack, based on an integral approach, is discussed in the next section. A low-Mach-number approximation suitable for inclusion in the finite difference equations is given in Sec. 3.8. While the utility of the TI - formulation at high Mach nurnbers can thus be veri-fied a~ high frequencies without recourse to experimental data, no such chèck is possible at low frequencies (Wo

<

1). The TI-results do seem to agree with experiment, but are otherwise open to question.

A clear weakness of the potential formulation is the assumption of uni-form entropy. Therefore one would expect the pressure and potential formulations to yield different results for heated jets. This does not appear to be t.he case, however. In the first place, whenever the excess temperature is sufficiently large so that velocity - induced refraction plays a minor role compared to temp-erature - i~duced refract~o~, results for the TI- formulation compare reasonably well with those for the p\l)formulation. Secondly, when the frequency is suffi-ciently high

(wa>

3) such

a

comparison can be made even when the velocity effect is as large as the temperature effect, /since it is then permissible to omit the gradient terms from the pressure formulation; again this comparison yields rea-sonable agreement.

By way of a summary the regimes of validity of the p(l) and TI- formu-lations are charted in some detail in Table 1. When the frequency is low and the Mach number high, the consistency of the TI- formulation with experimen~al

results (the agreement is best when the shear term is omitted) endorses the use of the TI-formulation for "engineering purposes", though leaving it theoretically suspect. It is likely that more reliable results than those yielded ~y the TI -formulation in this frequency - Mach number regime could be obtained only by simultaneous solution for the acoustic veloeities.

A tentative physical interpretation of the appare~t soundness of the quasi-potential formulation even at high subsonic Mach numbers is that ui" the component of sound velocity induce~ by the mean flow vorticity, is itself sol-enoidal. In that case the divergence of u.' in (3.19) vanishes and the simple wave equation without additional terms (3.3), rewritten in terms of TI, is

satisfied.

Finally it must be

poi~ted

out that the p(l) -and TI-formulations, though leading to the same results, are apparently at odds with experimental data for heated or cooled jets (see Fig.9). In all cases the computed tempera-ture effect is a good deal stronger than the measured.efflHet.cdlllre a.al1S1:l1o:li'lithe dis-crepancy may lie ifrthe convective upswing or droop (for heated and cooled hori-zontal jets respectively) which destroys the axisymmetry of the flow and may thus reduce refractive effects. On the other hand, the difficulty might stem from the neglect of heat conduction in the energy equation (3.8); if this is the case, the missing terms would have to be low-frequency terms, since finite-difference results at very high frequencies agree with ray-tracing results for heated jets (Fig.10).

3.3 Shear Term in Pressure Equation

The observation that the shear term appears to be a low-frequency term suggests that it is primarily associated with reflection or diffraction, rather than refraction. This follows from the fact that refraction effectB,

by themselves, are essentially frequency-independent. For example, a plane wave propagating through aplanar shear layer obeys the refractive law

sec~2 - sec~l

=

~

-

_M2

for a ratio of wavelength to shear layer thic~~ess ranging from zero(ray acoustics)

to infinity (see [11]). On the ot her hand, it is well known that acoustic

re-flection from aQ interface is strongest when changes in the properties of the medium are "abrupt", i.e., when the ratio of wavelength to the thickness of the

interface is large. Similarly diffractive effects are most pronounced at low

frequencies.

This reflection-diffrac~ion interpretation of the shear term is also

confirmed by the results below, which show that the shear term always enhances

the axial intensity. However, the shear term certainly does not account for all of the reflection and diffraction effects (although it might conceivably account

fully for reflection alone), since the wave equation without the shear term still

yields esse~tially. the correct frequency dependent directivity pattern over a wide range of frequencies.

It was suggested earlier that the shear term can be omitted from the

convected wave equation for the pressure at sufficiently high frequencies, so

that the rtumber of dependent variables are reduced from three to one ( in the

axially symmetric case). Such an approximation can be justified only by means of quantitative estimates revealing the ultimate effects of the shear term on //

the sound field in comparison with other terms. The following procedure provi~ès

rough estimates of this kind. /

Equation (3.15), for ~iform P, can be rearranged into

1 d 2p(1) d 2p(1) 2U d2p(1) dU dU U2 d2p(1)

+ 2p z

~

=

- 2 - 2 2

a dt2 dX.dX. a dX1dt dz dX_l a

~xl

l l

If all the terms on the RHS of (3.1q) are regarded as source terms,

that p( 1) must satisfy the integral equation

(1) 1

J [

2U P (~,t)

=

Pq +

47T

.

-7

i t d

3

y r

/

/

(3.26) is clear (3.27)

where

~

= t - ria, r

=

J x - y J and p is the pressure generated by a simple

source in a medium at rest (the corresp8nding source term, not shown in (3.26),

is a spatial 5 -fMllction wi~h a sinusoidal time-factor). This equation together with a second integral equation for u derived from the momentum equation, could by ~sed to obtain an iterative solutIon of (3.26).* Each new approximation to p\l) (x, t) would be obtained as the RHS of (3.27), using the latest estimates of pel) and u2 in the integrals. While this would be an extremely slow method

for solving the wave equation in the general case, it is relatively easy to carry

* The feasibility of such an integral-equation approach was originally suggested by H. S. Ribner

-/

I

out of the first iterative step when the Mach number is smalle In that case one may use the apalytical values of p\l) and u

2 for a simple souree in a quiescent

med-ium as the "zeroeth" approximation and substitute the corresponding derivates in the integra~ds of (3.27). The details of this procedure ~n~ the subsequent

nu-merical integrations to obtain a first approximation to p\l)(~,~) are given in

App. B.

Some of the results found by this method are shown in Fig. 11. The shear

term invariably co~tributes a very broad axial peak in intensity to the

direct-ivity patter~~at all frequencies and all distanees from the source. This

dind-nishes the depth of the refraction valley caused by the first integral in (3.27)

by about 1 dB for each tenth of the Mach number, irrespective of the frequency.

For À

>

2D this becomes too large a fraction of the overall valley dep~h to be

ignored. While the "critical wavelength" estimated in this way appears to be

about the same in the near field and far field of the source, it should be men-tioned that the estimates are probably "pessimistic" , particularly in the far

field, hecause the velocity distribution used in the integrations has a

rectan-gular transverse profile. For this model the wavelength is always long compared

to the thickness of the shear layer, so that low-frequency effects tend to be

exaggerated. ~he present conclusions, though based on a fi~st integral estimate

of uncertain acc~acy and a simplified flow model, are corroborated by the results

of the finite-difference approach described in Sec. 3.8. In both approaches, only

low Mach n~ers are considered. However, as both the shear term and the main

refraction term in the wave equation are proportional to Mj' the conclusions are

thought to apply equally well at higher Mach numbers.

3.4

Non-Dimensionalization and Transformation to Polar ,co-ordinates

The following transformations are based on (3.3), i~e., the shear term

and density gradient term have been omitted. Modifications for a hot jet and a

low Mach number approximation to the shear term are developed in Sec. 3.8.

In the steady state a sinusoidal source of frequency w will evidently

produce(a)sinusoidal pressure (or quasi-potential) field of the same frequency,

i.e., p 1 (or TI) can be written as B e- lwt where B contains the amplitude and.

phase information. Thus d/dt becomes -iw and multiplication of (3.3) by ~ e- lwt

gives 2 -w B ;;0 2 iwU dB +

if

dX l

2....2

- a V-B=O (3.28)

In terms of nondimensional distance: ~i

=

xi/D the dimensionless

quantities M

=

Uia and W

=

w D/a (which are functions of position in general)

this becomes, af ter multiplication by - n2/~,

+ 2i MW

_dÇ

dB ₊ W2B ₌ 0 (3.29)

Note that the three quantities Mj

=

u./aj' W

=

wD/_{a o} and C

=

a·/a toggther

with (3.29) and suitable boundary conaitlonsOfully determine B

Jfo~

a given

position of the source in the jet.

medium at rest suggests that the equation is most conveniently formulated in spherical polar co-ordinates rr,8,n such that

Sl rrcos8

S2 rrsin8cosn (3030)

s3 rrsin8sinn

with ojdn

=

0 owing to the axisymmetry of the source-jet configurationo The de-rivativès of B must then be written as

cm

~

o~

Os 2 1 cos8 oB ~-2 02B cos 8 -ocr2 +

! .

_rr Sln 2e sine OB

- -

_cr

_de

2 sin,8cos8

o~

+ -1 cr 2 ocroe cr 2 oB

""2

sinecos8

de

rr . 28

o~

Sln 082

=

2 1 . 8 [ 0 ~(cr sln8 ~) 2. oB +

d8

0 (sin8

d8)

oB

1

with

dn

0 cr Sln Consequently (3.19) becomes 2 2

o~

2' (l-M cos e) ~ + 2M sin8cose ocr +

(

2;M 2 sin2e + 2 _{i MW cose)} oB cr ~ +

(

1_2M 2_sin2_e rrtane - 2 i MW sin8 )

ä=d9

oB + W2 B 0 =0 (3.32)

The real and imaginary parts of this equation give two linear elliptic equa-tions involving the real and imaginary parts of B.

3.5 Finite Difference Grid

Various techniques can be used to find approximate solutions of bound-ary value problems [22J" For example, one might express the desired solution as a finite or infiI:\ite series of f\lnctions with unknown coefficients and then derive and solve sui~able equations for these coefficients, or a formulation in terms of an integral equation (as in App.B) could be sought. However, the method which has proved to be of most general usefulness and which was judged to be the most easily implemented for the present problem is the finite diff-erenee method, wherein a grid of closely spaeed intersecting lines is super-posed upon the solution domain and an approximate solution is computed at the grid points. At each grid point the approximate solution is required to satis-fy a finite difference equation, which represents an approximation to the

differential equation in terms of a locai- cluster of grid points.

A number of difficulties arise in the application of finite difference

methods to the present problem. First the domain of solution extends to infinity.

Therefore unless the last radial grid space is taken to be infinite, there are

infinitely many grid point equations. Secondly, the amplitude at a true poi~t

source with finite power output is infinite, and one cannot directly cope with infinite values in a numerical approach.

These two problems can be circumvented by confining the spatial grid

to the region lying between two source-centered spherical surfaces, one very

close to the source and the other far away. One must then stipulate approximate

boundary conditions on the two surfaces. Further discussion of this point is left to Section 3.9.

Now suppose one tries to solve the finite-difference analogue ~f (3.32)

at the lattice points of the finite grid. The points wi~l háve to be sufficiently

closely spaced to resolve the sinusoidal fluctuations in 'B; these fluctuations

are apparent if one writes B as the product of amplitude~d phase factor.

B

=

A ei<P

=

A cos<P + i Asin<P

The wavelength of the fluctuations is of order À, or about 2D at ,9000 hz for a

.75"

jet*. Assuming that about 10 points per cycleare required to describe

the fluctuations and that the grid terminates at 100 D , one must employ 500

circumferential grid lines. The required spacing of radial grid lines is more

difficult to ~ticipate. To assure proper equilibration of the matrix of the

finite-difference equations, however, it turns out that at least as ~ radial

grid lines are required as there are circumferential grid lines., If the grid

terminates much beyond 100 D, still more radial grid lines must be used, since

~he phase difference (an~ hence the number of cycles of B) between 8

=

0 and

8

=

~ grows like ~~cr (see Part II). Thus one arrives at a total of at least

2.5 x 105 grid points. At very high frequencies, for ex,mple 100,000 hz (near

ray-acoustic limit for

.75"

jet), this can become

3

x 10 or more. Thlis was

well out of the range of the 32,000 word core storage capacity of the available

IBM 7094 computer. Moreover, point or block iterative methods applied to the

linear problem diverge, because the matrix is not diagonallydomina~t, nor

amenable to a regular splitting (cf. Sec. 4.1).

Although B is sinuous and gives rise to the difficulties just

enumera-ted, ~he 'amplitude A and phase <P should vary mon~tonically with distance; if

A is written as x/cr and <P as crP, i.e., B

=

A-

_cr e

then

X

and P should be nearly constant. Thus by trading the real and imaginary

parts of B for the new depende~t variables

X

and P"a much coarser ~rid will

suffice to delineate the variables. The large st variations in

X

and P are

ex-pected to occur in and near the ini tial part of the j et flow', while at larger

cr ~ariations should subside. Thus the grid spacing may be allowed to increase with cr. However, instead of using variable grid spacing, it is simpler to transform the radial coordinate to

* Numerical values ha~e been chosen to allow easy compariso~ with the results