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Produced by the NASA Center for Aerospace Information (CASI)
a

TECHNICAL REPORT

Date: January 31, 1984

Submitted to: National Aeronautics and Space Administration

(Langley Research Center, Hampton, VA)

Title: ANALYSIS OF SOME ACOUSTICS-JET FLOW INTERACTION

PROBLEMS

Grant Number: NASA NSG1-113

Period Covered: April 1, 1981 - December 31, 1983

Submitted by: Pao-Liu Chow (Principal Investigator)

Professor

1617'8,g^ Department of Mathematics

Wayne State University

c^a

ti^^1

Detroit, Michigan 48202

l'^ Fig 1984+

(NASA -CR-175340) ANALYSIS OF SOME N84-16944

ACOUSTICS-JET FLOW INTERACTION PROBLEMS

Technical Report, 1 Apr. 1981 - 31 Dec. 1983

(Wayne State Univ.) 25 p HC A02/!!F A01 Unclas

CSCL 20A G3/71 00603

} 
SUMARY

The report is mainly concerned with the principal investigator's

research on some analytical problems in the interactions between the

mean-shear flows and the acoustic field in the planar and circular jets.

These problems are basic in understanding the effects of coherent large

structure on the generation and complications of sound in a sub-sonic jet.

Three problems have been investigated. The first problem pertains to a

spatial (vs. temporal) normal-mode analysis in a planar jet. This basic

problem is formulated as an eigenvalue-value problem. Since it is not of

Sturm-Lioville type, the wave numbers (eigen-values) behave in a much more

complex manner. This difficulty dictates the use of various methods of

approximation. Here we adopted an asymptotic method, known as the WYBJ 

method, Eventhough the method was designed for the high frequency case,

it is often found useful also for lower frequencies, as a result of analytic

continuation. In reality the disturbance consists o`_ many different frequencies.

Therfore one must deal with the wave packets. Given the spectral function

of the source, we showed how to determine the excited wave packet by the

Fourier-method and the method of stationary phase. In the second problem, 

we consider a slightly divergent, planar jet. Since, from a physical viewpoint

the exponential-growing wave violates the radiation condition. The unstable

waves must have a bounded envelope in space. In applied mathematics, this

defect is a manisfestation of a non-uniform perturbation analysis. To obtain

a uniform expansion, we recommend the two-variable method. By applying this

method, we derived the evelope equation for the excited waves, which is

amenable to numerical solution. Thereby the effects of divergent shear flow
Ey

7

on the sound amplitude can be assessed. Then we extend our analysis to a

parallel or non-parellel cylindrical jet. The third problem is concerned

with the acoustic waves in an axisymmetrical jet. By using the cylindrical

coordinates, we have shown that the analytical techniques used in treating

the planar jet, such as the WKBJ method, the method of stationary phase and

the two-variable method, can be applied to the parallel or non-parallel cylindrical

jets as well.

In the appendix, a recent paper entitled "Reconstruction of * Mutual

Coherence Functions for a Moving Source", is attached. This paper summarize

a related work partially supported by this grant, on determining the acoustic

source structure in a moving jet. The method is based the high dimensional

radon transform. Also it was shown that, in the case of line source, the

problem becomes similar to that in the X-ray tomagraphy.
ANALYSIS OF SOME ACOUSTICS-JET FLOW INTERACTION PROBLEMS*

Pao-Liu Chow

Department of Mathematics

Wayne State University

Detroit, Michigan 48202

Recently many workers in jet-noise research have investigated the role of

the shear mean-flow in the generation of sound. In supersonic flow, the modified

Lighthill's equations had been derived by Phillips [1], Lilley [2] and Pao [3],

to account for the shear-flow term in sound generation. The so-called shear

noise has also been discussed by Ribner [4], Mani [5] and others. All of

these studies are qualitative in nature. The experimental results of

Crow-Champagne [6] has stimulated a great deal of interest in the effects of

coherent shear structure on the acoustic amplification in the jet. In this

paper we shall be concerned with some basic problems in the acoustic-shear flow

interaction.

Perhaps, the simplest model for acoustics-flow interaction problem with

physical meaning is acoustic wave propagation in a two-dimensional parallel

flow of an inviscid, incompressible and non-conducting fluid. This problem

1

is similar to that of stability of two-dimensional parallel flows. A quick

.i

reading of the recent book on hydrodynamic stability (Chapter 4) by Drazin

and Reid [7] would reveal that, even in this simple situation, how little is

known about the general properties of its solution. This impression has motivated

us to re-examine the two-dimensional model. A number of authors had addressed to

the growth of spatial waves in parallel flows. However, due to radiation condition

at -, a spatially

* This work was supported by the NASA/Langley Research Center under the

grant NSG - 1330.
growing plane wave in a parallel flow seems physically meaningless. A good

discussion of this point may be found in [7]. It seems more reasonable to

account for the spatially growing modes by the divergence of shear flows.

This is the subject of several studies, e.g. [8] - [13].

This paper constitutes a summary of our results concerning some acousticsflow interaction problems. In section 1, the governing equations are presented:

Sections 2 and 3 are concerned with acoustic waves in a planar jet of constant

width. Section 4 pertains to a slowly divergent planar jet. Here we shall

introduce a two-variable expansion scheme to derive an evelope equation for

the spatial growth (or decay) due to the divergent flow. Finally, in Section 5,

we briefly discuss how to generalize our results for planar jets to the

axisymmetrical ones.

1. Governing Equations for the Mean Flow.

Consider the effect of mean flow on the sound propagation in an inviscid,

non-heat-conducting jet flow. Let U,u denote the mean and acoustic velocities;

p o ,p the mean and acoustic pressures and p o ,p the mean and perturbed densities,

respectively. By perturbing the Euler's equation, the continuity and the energy

equation and assuming that the mean pressure gradient terms are small, the

following set of acoustic equations may be obtained,

D + Yp o (v • u) = q (1.1)

Du I

Dt + o pp + (u - p) v = 0 p (1.2)

2
where Y - cp /cv is the specific heat ratio, q is an applied acoustic source,

and

Ft- (Tt + v • v) (1.3)

stands for the material derivative.

For convenience we use the notations U = (U,V,W), u = (u,v,w) and

i ^x,y,z) or (x,r , 8) , depending on the rectangular or cylindrical coordinates

are used. The following special cases will be considered

(A) Two dimensional jet: W = w = z = 0

(i) Parallel flow: V - 0, U - U(y)

(ii) Divergent flow: V = V(x,y), U = U(x,y) .

(B) Cylindrical jet: x = ( x,r,8)

(i) Parallel flow: V = W = 0, U = U(r) ,

(ii) Divergent flow: W = 0 U = U(x,r), V = V(x,r) .

In what follows, we shall give a more detailed description of Case A,

while Case B will be discussed only briefly.

2. Spatial Modal Analysis in a Planar Jet.

Let us first consider the most basic problem: the normal mode analysis in a

two-dimensional jet. In components, the governing equations (1.1) - (1.2)

with q -0 read

3
ti

i

F

[

bu

OF POOR Dp + y? (fix + ay ) ' q " 0

Du + 1 + vd- u— = 0 , -m<x<ao, a <y<b (2.1) P o ax dx

Dv + 

pi a 0

o

_ --

which is subject to appropriate boundary conditions,

where U (a + i') , U = U(y) (2.2)

According to the normal-mode stability analysis [7 ), we seek a solution of

(2.1) in the form for a < y < b

(p,u,v) _ @,u,v)(y)ei(kx-wt) (2.3)

For a given frequency w , we are interested in the dependence of the wave number

k on w . If k(w) is real, the mode is propagating or proper. Otherwise,

k is complex and the corresponding mode is non-propagating. If Im k Z 0 , the

mode decays and, hence, stable. However, if Im k <0 , the mode grows

exponentially as x-4 m . But this means, for fixed t, the modal amplitude will

grow along the downstream. It is physically meaningless to label such mode as

unstable, since it violates the boundary conditions at m . To account for wave

behavior, instead of simple mode (2.3), we shall later replace it by a wave

packet by superposing modes of different frequencies.

Let us substitute (2.3) into (2.1) to get a system of ordinary differential

equations in y

ikJ3 + Y p o (iku + ay) = 0

ikTu + J`' + ikp/p o = 0 , a < y < b (2.4)

ikTYv + 

p 1

dd7p = 0 ,

0

4

v
ORIGINAL PAGE

OF POOR QUALITY

where

I = U(y) - c, with c = (w/k) . (2.5)

After eliminating u,v in favor of p we obtain a second order equation

k211(1 - μ2 )^ 0, a<y<b (2.6)

N where

t

' P/P O

(2.7)

μ = 11/a , a = (ypo /p o ) 1/2 (2.8)

and

( )' = ddy

By a simple change of variable

(2.9)

the equation (2.6) may be rewritten as

L(k) k2(1 - μ2 )r - r* = 0 , a<y <b (2.10)

where

r

-`(11' - 211' 2 ) (2.11)

Here we assume 11 = U(y) -c # 0 in (a,b) . Otherwise we let c be complex

and define * as the principal -value as Im c-+ 0. Consider the homogeneous

BVP (boundary -value problem) associated with the original system (2.1)

L(k)* = 0 , (2.12)

Bl* = B2 * = 0 , (2.13)

where (2.13) stands for appropriate B.C.'s (boundary conditions). Let the

5
eigenfunctions *n with eigenvalues k , form a complete set in L2(a,b)

the space of square-integrable functions over (a,b) . Suppose we multiply

(2.12) by * , the complex conjugate of * , and integrate from a to b

to obtain

k2fb [1- (u) 2 11^1 2dy + 2k(W^fb UI^t12dy + J b

2 +[r(k)-(a) 2 11*1 2 } dy = 0 (2.I4)

a of

* a a

The difference between (2.14) and its conjugate yields

(k2 -k*2 )fb [1-(U—^)'11*(Y)I2dy +2(k-k*)fa(,)U(Y)I*(Y)12dy = 0 (2.15)

a

The above show that the eigenvalues need not be real, even though L(k) is selfadjoint. Note that (2.14) may be written as

bok2 + 2b 1k + b 2 = 0 , (2.16)

where'. o , b l , b 2 are the corresponding coefficients in (2.14). Thus, if r is

independent of k , corresponding to each eigenfunction^ n , there are two eigenvalues

k which are roots of (2.16) . Let

(2.17)

be the discriminant for (2.16). Thus, if p (*) z 0 , the roots are real. The

condition for n-th mode being proper at a fixed w is that

Agn ) x0 ,

otherwise k are complex so that this mode either grows or decays in x .

As a simple example, consider the BVP

6

,a--
OF POOR QUAI L(k)* = 0, -b = a<y<b

OF P ^ (2.18)

*(-b) (b) - 0

where U is constant. Then L ( k)* - ,U" -k2 (1 - 112 ) * and the eigenfunctiaons are

(Sin 2b ( y+b)) , corresponding to eigenvalues kn determined by (2.14)

(1 - (at ) 2 ]k2 + 2( 2)wk + [(2b) 2 - (a) 2 ] - 0 (2.19)

C1

for which

4(^n) R (a) 2 - [1 - (a) Z l(b)2 2 , n

Thus, for a fixed w , there are a finite number of proper modes k with nit N ,

where N is the largest integer less or equal to 2bm 2 with m - a

WT( - m ) 1

The rest of the modes are improper and may grow or decay.

Next, suppose the velocity profile U(y) is symmetric and varies smoothly

over [-b,b] . As shown by (2.19), k - 0(n) for large n , the behavior of

higher-order modes may be determined by an asymptotic method. The well-known

WKBJ method can be applied. Let

* — A(y)ek6(y) as k-4 a , (2.20)

where A , 6 are determined by the d.e. (2.10). The general solution can then be

written as

(1-μ2) -

1/4(C Sink 0 +C

2

 Cos h kA)

where

8 = fy 11 - μ2 (s)] 1/gds .

a

The approximate solution is valid only when

μ = a-1 (u - c) 0 ±1 , a<y<b .

The points yo at which μ(yo ) - +1 are called turning points. In th

7
ORIGINAL PAGE 18

OF POOR QUALITY

the solution ( 2.21) is not valid near y - y o , and a different form of solution

must be constructed. Introduce an appropriate change of variables (14]

9 - f ( y ) , C - 8(Y) Cy ) , (2.24)

the equation (2.3) yields

d92!; - k2^{ (2.25)

where

f - (3 1Y(1-μ2)1/2dy,]2/3

a

(2.26)

1 f'" 3 a= - r

c -

2 

(f') 

3 4 (f , )4

(fl) 

2

For large k , the equation ( 2.25) is approximately an Airy equation to that

C... CA (g) + dBi (g) as Jkl -P w (2.27)

where Ai , Bi are Airy functions of first and second kind. Hence the solution to

(2.3) is given by, referring to (2.26)

*04 — W WI (f(y)] + d B i 1f(y)1) . (2.28)

Suppose that the condition (2.23) holds. Then, in view of (2.21), the

eigenvalue problem ( 2.18) with non-constant U has the asymptotic solutio ►

^n (1-μn)-1/4S in kn jYb (μn(s)-1] 1/gds, for large n (2.29)

where μn My) - w/kn ] , and k are solutions of the equation

2

- +nrr 1 . (2.30) fbb ( k2 ( 1 _ U 2v)] + 2k ^2)U(Y) - (a)2 }1/2dY

8
We note that (2.30) reduces to (2.19) when U is constant. For large k , Jk) >> w

(2.30) has no real solutions. Thus all higher modes are improper for a finite

W . However, if W is so large that W >> jkj , (2.30) gives

b

f (W-kU(y))dy••. ±nTrcr

-b

b

k (W - (2b )) 2b ^b U (y ) dy

or

(2.31)

which is valid if W — (ET-TA ) and n is large. This value of k gives rise to

(2bU )x + i U' (x - U0

 t)

p 

— ^ 

n 

e

o 0

'

(2.32)

b

where Uo - 1 Sb U(y)dy

assumed, by continuation,

(2.32) shows that at high

with the propagation sppei

is the mean-velocity of the flow. Here we have tacitly

(2,.30) holds also for smaller values of k. The result

frequency, the disturbance is a spatially modulated wave

i U 0 . On the other hand, if m« `k) , (2.30) has the

approximation

k f (1 - (U)2)112dy + A2 )^ b l- U 2 1/2 dy + a r i , (2.33)

or

b

( 1 -

2 ) 1)2

corresponds to the non-propagating modes.

In the presence of a turning point, we should apply the B.C.'s to ( 2.28).

But the resulting characteristic equation is not readily solvable to get

explicit results as before.

We remark that, for simplicity, we impose the B . C. p -0 at y - +b .

More realistically the field in jyj < b must be matched to th + t in jyj > b

at y - +b . This correction is straight - forward but complicates the characteristic

equation whose explicit solution becomes inaccessible.

ORIGNAL

OF POORQUAI! i Y 9
ORIGINAL PaGL IN

OF POOR QUALITY

Instead of simple modes, consider the st- perpositiun of modes excited by

various frequencies

a

P( t ,x,Y) Fl pn ( t , x ,Y) , (2.34)

where

i{k (w)x -Wt}

pn (t,x,Y) "'^;n(Y, kn , cu ) *n(Y,kn.w)gn(u^)e

n

dw (2.35)

is the n-th mode of p , and k n (W) is the n-th eigenvalue of the BVP (2.12) - (2.13)

corresponding to the frequency W . It is easy to verify that p given by (2.35)

satisfies ( 2.1) after u,v are eliminated. The solution (2,.34) may describe the

disturbance due to a source located at x -0 .

Since we are interested in the event near the wave front in the large

distance down- stream, the asymptotic evaluation of (2.33) will be carried out as

x .a m , t -# s with (x/t) fixed. Let

t/x - T . (2.36)

Then (2.33) may be •written as

P

n 

=% ^n(Y,kn,W)e

n

dw , (2.37)

where

% - 1L (Y,kn )*ngn . (2.38)

Hence, as X-4 • , we may apply the method of stationary phase for asymptotic

evaluation. The stationary points of the phase are determined by the equation

k' (w) t (2.39)

Let W* be a solution. Then the contributiot, to p due to this stationary

point is (15)

10
Of POOR QUALITY

p ( k*,w*) [ 2TT 1/2exp i (k*x - w*t - n sdn V (w*) ) ( 2.40)

n n n x kn (w*) n 4 n

where kn = kn (w*) .

For example, we consider the case of constant U . By (2.19), we have

It = -M(2) ± en/2 

, An = (a ) 2 - (1-M2 )(b) 2 2

(2.41)

where M = Vat is the Mach number. It follows that

kn (w) _ - (M) ± 2w1/2 (2.42)

ad

n

k"n (w) _ + CV-2 A -1/2 (1- 2 —) (2.43)

Cr 

An

Then the equation (2.39) has the solutions

wn = +a( aT +M) (1 - M2 ) ( b) [1 + (Off +M) 2 1

1/2 (2.44)

so,

kn = -wn {a ± a(ar +M) } with T = t /x (2.45)

Thus we see that, due to boundary absorption, each mode excited by a source

at x =0 will evolve into space-time modulated waves which decay like IxI -1/2

downstream. Their respective speeds, referring to (2.42).and (2.43) are given by

C = Iw*/kn l = ,M + (M + at/x) -1 ) a . (2.46)

For a variable profile U , if the variation is slow, the WKB method can

be shown to apply. Thus, if the dispersion equation (2.30) may be solved

approximately, a similar procedure can be applied to give an asymptotic

evaluation.

11
ORIGINAL PAGE 19

3. Radiation in an Unbounded Planar Jet. OF POOR QUALITY

In the presence of external source, the RHS of the first equation in (2.1)

1

is q(t,x,y) 0 0 . Let

` f

1 2 rr f(t,x)e-i(kx-wt)dxdt (3.1)

(2TT) JJ

i

Then a Fourier transform of (2.1) with q 0 0 yields (2.4) with the RHS of the

first equation equal to q # 0 The eliminated system corresponding to (2.10)

reads

L(k)* = ^^^ _ k2 (1-μ2 ) * - r* = -ikq < y< (3.2)

t Rte = 0 at y = ± cc (3.3)

where the B.C. is introduced by the radiation condition for p . Suppose

G(y,y',k,w) denotes the Green's function for the BVP (3.2), (3.3). Then

we have

j(y,k,w) = f(-ik)G(y,y',k,w)q(y',k,w)dy' (3.4)

It follows that

p( t , x ,y) = (iPo)-1JSSkJ(y,k,w)G(y,y',k,w)

(3.5)

X q(y',k,w)e

i(kx-wt) dy' dkdw .

By the theory of contour integration, the major contribution to the integral

(3.5) comes from the singularity set of G Let *142 be two complementary

solutions of (3.2) satisfying the B.C.'s at y = m,-^ respectively. Then we

have

1(y')*1(Y)*1 (y ') y Z y'

G = (3.6)

-A (y')*2(y)*i(y') y s y'

12
ORIGINAL PA 'G'S. 77

OF POOR QUALITY

where A - (*1*2 - ^2^1) is the Wronskian which should be a constant.

As an example, suppose q is a point source at x - 0, y - 0 , so that

q = g0 (t)b(x)b(y) . Then (3.5) becomes

P( l , x ,Y) = (ipl-1 fkj(Y,k,w)G(Y,o,k,w)go(w)

(3.7)

X ei(kx-wt)dkdw

For a slowly varying velocity U at low Mach number, the result (2.20) by WKB

method gives

G(y,o,k,w) = G(-y,o,k,w)

(3.8)

1

go 1/4 ik:yg(s)ds

o

2ikgo [g(y)p

e , Y 2 0

where g(y) = [μ2(y)-1)1/2 , g o = g(o) . Upon using (3.8) in (3.7), it yields

1 -1 go(k,w) 1/2

P(t , x ,Y) (-Po ) ff go (k,w)^1(Y,k,w) $(Y,k,w) qo(w)

i kf Yg (s,k,w) d s +i(kx-wt)

x e o dujdk, Y a

It is difficult to compute the integral in closed form. Sometimes

possible to evaluate it approximately.

13
OF POUR QUALrry

4. Propagation in a Slowly Divergent Planar Jet.

Let the mean-flow be divergent so that, in (1.1), (1.2),

U (U,V '0) ,

(4.1)

u = (u,v,o) ,

where U,V are functions of x,y . As a function of x , they are slowly varying.

At low Mach number, the mean flow is assumed to be incompressible so that there

exists a stream function ^(g,y) , where ex is a slow variable with small

e > 0 . By definition we have

U ^y(g,y) a ^2(t'y)

(4.2)

In terms of the governing equations ( 1.1) - (1.2) can be written in 

components as follows

11 ll _ 1

Deu + (sY 12u + p22v) 

a

Po x

Dev - (e 2^ u + e^ 12v) -

1

(4.3)

Po by

r)u av

De P +ypo ( --+ sy) =q

where De 11

( at

a

+ Y2

^

1

ax - eY 1 a

a

y)

When the skew parameter e = 0 , the system (4.3) reduces to the parallel flow

case (2.1). Here we wish to determine the effect of divergence on the

propagating waves in the jet. By the two - variable method (16], we write

F = (p,u,v) .

14
Then (4.3) may be put in the form

ORIGINAL PAGe:; 1j;

OF POOR 01j,4- iY

(4.9) c)F = LZ 1Z

bX ax + e 69

Expand Z in a power series in a to get

Z(X,g,e) - Z 0 (x,D +

In view of (4.8) - (4.10), (4.7) yields th,

15

D`F + BeF - Q ,

where

o a

o bX Ypo by

= 1 a

Be e

po ax Y12 ^22

1

po ^y a ^11

_ 

412

q

Q 0

0

First consider the homogeneous B.V.P. in free-space

De F + Be F = 0 ,

on which we impose the radiation conditions at m .

Surpressing the dependence on t and y , let

F - Z(x,g,e), g - ex

so that

(4.4)

0.5)

(4.6)

(4.7)

(4.8)
ORIGI14AL N :max'::; r

OF POOR QUALM.

(4.11)

(4.12)

(D +B)z0 = 0 ,

(D+B)Zl = Ql n -(D1+B1)ZO

where

D=DO s (catYax) +

s

D a -

1 2Ti 1 a^y

0 a

YPo ax

^

YPo by

1 .SZ B

Po ax 0

1

Y22

la

_ 0 o by

0 0

o Ypo s

a 

a o

Bl = i a

00 at

1

Y12

0

0 0

412

1

=1

I (4.13)

Note that ( 4.11) is the parallel flow problem (2.1). To proceed we consider

the adjoint problem to (4.11)

(D* + B*)F* - 0

where D* and B* are adjoints to D and B , respectively. Now,

be a solution of (4.11). Then so is

Z0 (x, g ) = a(t)FO(x)

where CI(g) is as yet to be determined. A substitution of (4.15) i

results in

16
(D+B)Z1 - -(B 1 +D1 )a(g)Fo (x) . (4.16)

Let F,G be two vector-functions of t,x,y . Then we define

(F,G) - frf (F•G)dtdxdy ,

where F•G is their dot product. In view of (4.14), we have

(F*,(D+B)Z1) - ((D*+B*)F*,Z1) - 0 . (4.17)

Thus (4.16) implies that

(F*,(B1 + D 1 )F0)Q(g) - 0 (4.18)

which, noting (4.13), may be written in the form

or(g) a + 9 (067 = 0 . (4.19)

This is the envelope equation describing the slow modulation of waves due to the

divergence effect.

Alternatively the problem can be treated by first applying a Fourier transform

to the systems (4.11) and (4.12). The resulting equations may be reduced to scalar

equations for the pressure field as done before. This is particularly convenient

when dealing with the radiation problem (q 16 0) .

The evelope equation (4.19) is difficult to solve analytically. But it is

amenable to numerical treatment. For example, in the study of the boundary layer

stability, Saric and Nayfeh [17] applied two-variable method to account for the

slow divergent flow. They derived an evelope equation similar to (4.19). Their

numerical procedures for solution is applicable to the preset

17
• f

ORIGi"tiHL FAGS 13

OF POOR QUALITY

5. Acoustic Waves in an Axisymmetrical Jet.

The analytical techniques for planar jets may be generalized to an

axisymmetrical jet. Introduce the cylindrical coordinates (x,r,A) , with the

corresponding velocity components ( U,V,W) and ( u,v,w) .

First consider the cylindrical jet, i . e. U - U(r), V - W n 0 . In

components, the system ( 1.1) - (1.2) reads

Dp + ypo

(

cox + r ar(rv) + r c18)

q

Du+l 

ax 

+ 1 va= ( rU) -0 Po

Dv + —^ (rp) 0

Por

Dw+i 

1 ^

=0 Po

(5.1)

where D = ( a + U-X)

For the homogeneous case, q B 0 , we set 'w

(p,u,v,w) _ (p,u,'v,w)ei(kx + mg-u^t) (5.2)

Then (5.1) is reduced to

a

ikTrp + ypo {iku + _ = (rv) + it w) 0

- a

S

ikTru + (ik/po )p + v(r ) = 0

(5.3)

ikTjv + ( 1 /p

o

)= ar(rp) = 0

ti=

ikTw + (im/po r)p - 0

After eliminating u,v,w, in favor of p , the system ( 5.3) yields
(MIGENAL PAGE 13

OF POOR QUALITY

2

Mm ( k)^ - TID

)lP = 0 r (I Dr a(r)DJ I )P - m2 + k2 (1-μ2 (5.4)

r

where

Drf = _ -r(rf), a(r) - (DrV/T1 (5.5)

The solutions of (5.4) for m = 0,1, known as the axisymmetric and helical modes,

are of special interest. In contrast with ( 2.10), Eq. (5.4) is naturally associated

with an equation of Bessel ' s type. This is true when U is constant. Then (5.4)

becomes a Bessel-like equation, but not exactly.

It is possible to generalize the analytical results in 12 to the present cabe.

By (5.5), the equation ( 5.4) may be written as

p" + (r - 2T1 - 1J' )P , -

{

m2Z1 + k2 ( 1 - μ2 ) + }p 0 (5.6)

r

Set

p - N (5.7)

in (5.6) to get

2

Lm

 (k)* +1 - {k2 (1-u2 ) + m 21 )* + g(r)* - 0 (5.8)

r

where

g(r)- (

11=

^- 2 - L) . (5.9)

For instance, for large k , we can apply the WKB method or a two-variable

i

expansion. The procedure is similar to what was described in S2. One possible

approach is to seek a solution of the form

= a(r)H(l)(r) or 0(r)H(2)(r). (5.10)

where X - A2+1 is the order of the Henkel's functions of first or second

kind. By substituting (5.10) into (5.8), we may derive equations for a or

to which we apply the asymptotic method.

19
OF HOCR QUAL17'Y

For a divergent jet, the mean flow has the velocity components

U- 1_ (rY) , v - - JL Y , w a 0 , (5.11)

where J(g,r), g = ex , is the slowly-varying stream function. The governing

equations ( 1.1) - (1.2) for this case takes the form

De P + YPo (T x + Dry + r a A ) - q .

1

0

_

De u + P

o

1 

ax + 412u + ^22v '

(5.12)

De v + P1 Drp - e 2+ llu - e^12 v - 0

Po

D w + 1 lap-0

C Po r d A

where

a _ 1

Ds - ( clt + ^2 cox c Y,Dr ) , (5.13)

and

Yl -

A' Y2 - Dr^. and so on.

Again we introduce two variables x and g - ex . Then a two-variable expansion

procedure as presented in the previous section may be applied to obtain an

approximate solution to the homogeneous form of (5.12), (q 2 0) .

20
References

1. Phillips, O.M., J. Fluid Mech., 9 (1960), p.l.

2. Lilley, G.M., Tech. Report, British Aeronaut. Res. Council, Sept. 1958.

3. Pao, S.P., J. Sound. Vibr., 19 (1967), p.401.

4. Ribner, H.S., J. Sound. Vibr., 52 (1977), p.121.

5. Mani, R., J. Fluid Mech., 73 (1976), p.753.

6. Crow, S.C. and F.H. Champagne, J. Fluid Mech., 48, (1971), p..547.

7. Drazin, P.G. and W.H. Reid, Hydrodynamics Stability, Cambridge Univ. Press (1981).

8. Chan, Y.Y., J. Fluid Mech., 17 (1974), p.46; p.1667.

9. Hardin, J.C., AIAA 7th Aeroacoustics Conf., Oct. 1981.

10. Crighton, D.G. and M. Gaster, J. Fluid Mech., 77, (1976), p.397.

11. Moore, C.J., J. Fluid Mech., 80, (1977), p.321.

12. Plaschko, P., J. Fluid Mech., 92, (1979), p.209.

13. Bayliss, A. and L. Maestrello, AIAA Journal, 19, (1981), p.835.

14. Erdelyi, A., Asymptotic Expansions, Dover Pub. (1956).

15. DeBruijn, Asymptotic Methods, Dover Pub. (1958).

16. Cole, J.D. and J. Kevorkian, Perturbation Methods in Applied Mathematics,

Springer-Verlag (1981).

17. Saric, W.S. and A.H. Nayfeh, Phys. Fluid, 18(1975), p. 945.

21
APPENDIX

"Reconstruction of the Mutual Coherence Function for a Moving Source"

SIAM Journal on Applied Mathematics, 43(1983) pp. 1439-1445.

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