Numerical solutions of acoustic wave propagation problems using

NASA Contractor Report 172434 ICASE REPORT NO. 84-39

NASA-CR-172434

19840025041

ICASE

_. _.2_(L

....

....

: :- _'-'_" "_

s' -j

NUMERICAL SOLUITONS PROPAGATION PROBLEMS COMPUTATIONS

S.

WAVE

I. Hariharan

Contract August

INSTITUTE NASA

OF ACOUSTIC USING EULER

No. 1984

FOR COMPUTER

Langley

Operated

NASI-17070

Research

APPLICATIONS Center,

by the Universities

IN SCIENCE

Hampton,

Virginia

Space Research

AND ENGINEERING 23665

Association

National Aeronautics and Space Administration

LANGLEYRESEARC_ ,3kr'_TER LIBRARY,r_IASA

Langley Research Center Hampton.Virginia 23665

HA,_,_PTON, VIRGINIA

NUMERICAL

SOLUTIONS

PROBLEMS

OF ACOUSTIC

USING EULER

WAVE PROPAGATION

COMPUTATIONS

S. I. Hariharan* University of Tennessee Space Institute Tullahoma, Tennessee 37388 and Institute for Computer Applications in Science and Engineering NASA Langley Research Center, Hampton, VA 23665

Abstract This paper reports solution procedures for problems arising from the study of engine inlet wave propagation. The first problem is the study of sound waves radiated from cylindrical inlets. The second one is a quasi-one-dimensional problem to study the effect of nonlinearities and the third one is the study of nonlinearities in two dimensions. In all three problems Euler computations are done with a fourth-order explicit scheme. For the first problem results are shown in agreement with experimental data and for the second problem comparisons are made with an existing asymptotic theory. The third problem is part of an ongoing work and preliminary results are presented for this case.

Research was supported by the National Aeronautics and Space Administration under NASA Contract No. NASI-17070 while the author was in residence at ICASE, NASA Langley Research Center, Hampton, VA 23665.

i

[

I.

Introduction A considerable amount of work has been done in the past to

understand acoustic wave propagation problems.

They occur

in

several situations such as sound radiation from engine inlets, exhausts

and

underwater

acoustics.

In

this

paper

we

are

primarily interested in the problems arising from engine inlets, but

the

methods

we

develop

here

underwater acoustics problems too. two parts.

nonlinearities.

applicability

to

These type of problems have

Both have complex wave structures due to

One likes to calculate the sound pressure field

inside the

accomplished

the

The first one is the inlet wave propagation and the

other is the radiation.

both

have

by

inlet

and

solving

in the

both

atmosphere.

parts

together,

This can which

be

becomes

computationally complex, or by a coupling procedure by solving in the inlet and then in the atmosphere separately. A great deal of engineering literature exists on this subject, in particular for linear problems.

This paper is intended to report a sequence of

successes of Euler

computations of both

acoustic wave propagation from inlets. natural

way

of

doing

these

linear and nonlinear

It turns out this is a

calculations,

since

the

field

equations are obtained from Euler equations. The difficulties in this approach are mostly attributable to treatment of boundary conditions. arise

when

numerically.

one

wants

to

Particularly, the difficulties

prescribe

boundary

conditions

It is known in linear wave propagation problems

that it is difficult to prescribe farfield boundary conditions

-2-

numerically.

The

Sommerfeld's in the

radiation

far

outgoing

corresponding

field.

waves

conditions.

holds

frequency

the

simulated

by

several

solve a

equation

acoustic

similar

which

in the in

a

near

the

in reference follows

the

envelope

method.

I

the

reference

2

appropriate.

family

the

for

of

is

the

radiation

problem

discussed

linear the

but

we

A

wave

radiation

from

III.

This

solved

the

problems

are

inlet

solutions

subject

sound

does

the not

reflect

is to examine

subject

to area

sound

parameters

be

at

if one

variation

source can

and

the can of

strength. used

to

open

end

attenuate the

inlet, A

reduce

we

to

of

mean

flow

combination noise

from

we

the

and

that

in

goal

section

number

these

engines

the

The

exit

Mach of

which

considered

fact

the

the

in

inlet

inlet.

at

of

from

inlets

the

the

sound

work

implemented

the

concerned,

in

is called

considered

inside

nonlinear

as

is

of

idea

method

work

and

boundary

the

cylindrical

As

combine

version

and

domain

this

simultaneously.

far

to

be

farfield

pattern

problem

can

is another

time

problem

frequency

exact

first-order of

in

nonlocal

There

conditions

sound

of

atmosphere

of

here

of

in Section

context,

models

the

seems

In

as

is

An

the

higher-order

in the

field. 7

9.

of

of

method

context

reflections

conditions

is obtained

scheme

wave

well

such

numerical

the

as

are

behavior

family

farfield

One

which

no

asymptotic

domain

domain

solutions

equations

the

time

methods.

is available

the

2

conditions

guarantees

extract

in

procedure

conditions

to

This In

which

reference

used

domain.

integral

conditions

In was

asymptotic

and

and three these

-3-

facts In

were

demonstrated

this

case

problem.

We

section

of

author no

we

is not

class

of

the

linearized

variable

nonlinear

condition.

be

Such

and

obtain

on

the

are

consider

model

in

Section

V

dictate

of

will

the

procedure V

are

the

and

end

due

an

In

incoming

approximate

to

specifying

IV

a

strictly

to

outgoing

the

obtain

for

inlet.

incoming

Section

a

exit

that

equivalent in

the and

IV and

open

to

becomes

at

nonlinear

The

open

zero

conditions

waves

end.

can

We and

open

the

condition.

dimensional

acoustic

near

to

a procedure

boundary

in Sections

one

set

of

experimentally.

condition

the

equation

simply

decades

equations

discussed

variables

will

impedance

any

equations

characteristic

of

field

at

two

nature

The

model

problems of

last

different

of

our

the

reflection

inlet.

for

linearization these

a no

aware

reflection

this

face

desire

the

in

an

quasi-one-

two-dimensional

model. In

all

three

fourth-order section. that

we

accurate This

scheme

tried.

In

(Section

IV),

we

became

tried

the

Chebyshev

inlet the

and

essential

order

situation method

a

the

the

is superior

Chebyshev

This

for

this

the

other

next

methods

near

the

end

at

the

we

class

the

fact

points

inaccurate

center but

model

that

to

were

it

believe

a

that

due

calculations

Hence,

in

used

polynomials

was

an alternative,

implement.

than

we

quasi-one-dimensional

with

are

here

discussed

results

concentrated

nonlinearities

to

for

expensive.

is

is

better

method

are

discussed

which

provided

result

technique

problems

scheme

spectral very

a

of

particular

points

as

Multidomain our

classes

of

the

of

because inlet.

is cumbersome that

of problems.

the

the

in

fourth-

-4-

The plan of the paper is as follows.

In Section II we give a

description of the fourth-order method and its usage in higher dimensions.

Sections III, IV, and V contain the description of

each problem under consideration.

Finally,

in Section VI we

present some numerical calculations. We omit derivations of the field equations that we solve in our discussions and we refer readers to references 4 and 5.

The

derivation of field equations of the problem V will be reported in a forthcoming paper.

II.

Numerical Scheme As stated in the introduction the scheme used here is an

extended

version

reference 3.

of

MacCormack's method

and

is developed

in

To provide a brief description let us consider a

single equation of the form

ut + fx = h.

Let

xj (i ( j (N)

domain.

(2.1)

be equally spaced nodes in the computational

Define forward and backward flux difference operators by

P_.(f) = 7f. 3 3 - 8fjeI + fj£2" Then the scheme has four steps.

From a time

nat

it has a backward predictor and a forward corrector

(2.2) to

(n+_2)At

-5-

(i) = un ap-(fn) + 8h(n) uj J 3 J

(2.3) j Un+I/2= 1/2 Uj +

In the next

At/2

j

+ aP

f

+ 8h

.

time-step it is changed to a forward predictor

and a backward corrector stage as follows:

u!l) = un+ 1/2 + ap+ fn+ 1/2 + Bhn+ i/_

J

J

J

3

(2.4)

where

a = At/6Ax, 8 = At/2

and

the

superscript

(i) denotes

predicted values. The other subscripts have the usual meaning of values evaluated at the indicated time step.

From (2.2) one must

notice that the fluxes are not defined at the end points, namely at

j = 0,i and at

extrapolations.

j = N+I, N+2.

There one can use suitable

In all problems discussed here we used a third-

order formula as follows:

fj = 4fj+l - 6fj+2 + 4fj+3 - fj+4'

(j =0, -i) (2.5)

fj+l = 4fj - 6fj_1 + 4fj_2 - fj-3'

(J = N, N+I)

-6-

In

more

technique.

than

one

dimension

we

use

operator

Let us consider the following two-dimensional system:

w t + --x F + --y G = -H.

If

splitting

Lx(At/2) and Ly(At/2)

(2.6)

denote symbolic solution operators to

the one-dimensional equations

W t + F x : H1 (2.7)

w t + Gy = H2.,

in which

H I and H 2

are suitable decomposition of

H,

then we

solve (2.6) by n+ 1/2 w

n = L x Ly w

(2.8) wn+l = L L wn+ 1/2 -y x --

It must be noted that the operators form for each

At/2

above splitting

Lx

and

Ly

change their

interval according to (2.3) and (2.4). The

is known to preserve second-order accuracy in

time and does not change the spatial accuracy of the scheme. The stability criteria is chosen in such a way that it is common for both equations in (2.7). The

last

aspect

artificial viscosity

we

consider

here

is

to resolve shock waves

the

addition

of

for the problems

-7-

considered

in

conservation

Section forms

of

IV. the

The

equations

considered

there

are

nature

w t + f(w) x = 0.

The

artificial

is of

the

where

viscous

of

term

added

to

(2.9)

(2.9)

is of

second-order

and

form

_ = 0(i)

and

p

is the density.

The difference form of

this is

" _-_ IPj+1 - pjll_j+l - _j)

I Pj - Pj-II(_j - _j-l)] " This is a second-order formula.

Thus we settle for less accuracy

in the presence of shocks to obtain sharper shocks.

III. Problem of Sound Radiation From Unflanged Cylindrical Inlets The problem studied in reference 4 is presented here.

The

same problem has been studied experimentally15 and analytically using

Wiener-Hopf techniques.13

radiated sound

in the atmosphere

pressure field inside the inlet.

The goal here is to calculate subject to a

given acoustic

Euler computations for this

-8-

problem were done only in the absence of mean flow.

With mean

flow, which simulates flight situations, results are available in reference 7.

Here

we present

the

equations and

approximate

boundary conditions for sound radiation for an incident spin mode 'm'

(m)0)

of sound pressure wave at the left end of the inlet

which is cylindrical (see Figure i).

The field equations in this

case are _p + v + imw_ 8--_ + Uz + Vr r

0

Du 8P = 0 "FE + %-£ (3.i) _v _p _ %-£ + %-f - 0

8w im 8--_ + _---p = 0

where

(u,v,w) are components of acoustic velocity in

coordinates and case

m = 0

p

and

is the acoustic pressure.

(z,r,8)

For the plane wave

w = 0.

The problem here then is to solve the system (2.1) subject to an incident field of the form

i£mZ p " e

together with hard

ikt JmIlm(r/a))e

(3.2)

wall conditions on the inlet (duct) wall and

radiation condition in the atmosphere. zeroes of the Bessell function

J'(z) m

In (3.2) and

a

lm

are the

is the radius of

-9-

the

duct

and

£m

are

related

to

the

£m = /(ka)2

By

considering

at

the

inflow

left

the end

of

reflected the

8v

and

by

(3 3)

"

travelling

(3.1)

ka

in the

(3.2)

give

-z

direction

the

following

wall

of

z (r)t

u t = -2ike

m

Jm

Im

e

(3.4)

Im J'mIlm (r/a) 1

8--t+

the

12 m

number

conditions

P + _m

On

-

waves

inlet

wave

the

a JmIlm(r/a)

] '

p : 0.

(3.5)

inlet 8P 8r - 0

which the mode

is axis

the

standard

takes

number.

rigid

different

They

wall

forms

(3 6)

condition. depending

on

The

conditions

the

incident

on spin

are

v = 0 v + iw = 0 v = O, w = 0

on r = 0 on r = 0 on r = 0

(m = O) (m I) (m) 2).

(3.7)

-i0-

FarFieldBoundary

73

72

Directivity Measurement

" ,, "

_'1

× ! !

i

\

D I nflow Boundary

Semi-Infinite Inlet s

s

i_,,

i | \i

_

i, z

fs

t

Y

Figure 1

In

the

one

atmosphere

needs

distances. of

such

we

this

circle

of

of

apply

for such

domain

at

Then

duct

R

from

and

the

conditions

of

reference

2 at

71,

and

73.

This

72

situation

to

be

several

problems

each

larger

domain

one

origin

(which

each

point

can

condition

and the

can

one

these

in

needs

the

Figure

taken

far

families

construct is

line) of

on

at

presented

than

See

point

center

are

conditions

distances

Thus

imposed

introduction

model

at

the

on

to

reflection.

measurements.

computational

radius

without

boundary

directivity

purpose.

the

in the

conditions To

of

radiates condition

mentioned

desires

boundaries

end

As

2.

one

for

nonreflective

computational

where

sound

a

boundary

reference finite

the

region 1

again

farfield locally

at

apply

the

u +

v

sin

u)

+ p

a

open

radiation

boundaries be

shown

for

this

be

_--_SP - (u cos

a

=

0,

(3.8)

-ii-

where

a

is the angle measured from the z-axis to a point on

YI' Y2 or Y3" With the above formulation and with the scheme given in the last section the solution was achieve

a

supplied

steady

from

started at a state of rest to

time-harmonic

the

state.

inflow conditions.

Time The

increments

solution

are

for this

problem will approach the solution of Helmholtz's equation times eikt.

(It is immediate from (3.1) upon taking Fourier transform

with respect to time that one obtains the Helmholtz equation.) The

philosophy

mathematicians

behind

this

the

limiting

as

procedure

is

amplitude

known

among

principle.

The

advantage of doing this problems in time domain is that one does not

encounter

the

problem

of

resolving

or

calculating

the

interior eigenvalues in the inlet.

IV.

Quasi-One-Dimensional Model In this section we describe a simplified nonlinear situation

which

has

been

phenomena.

popular

in

analyzing

the

nonlinear

wave

We summarize the work appearing in references 5 and

6. This model is depicted in Figure 2.

The duct corresponds to

the inlet situation and it has a constriction at the center. There

is

a

steady

flow

from

right

to

left

propagates from a source upstream of this flow.

and

the

The derivation

of field equations used here are available in reference 5. are

sound

They

-12-

qt + lUsq + qsu + Uq)x = 0

ut + where u

and

UsU +

q = Ap p

quantities

with

+ Cs

(4.1)

--_--qq

A = A(x)

x = 0,

in the area variation of the duct

are acoustic velocity and density respectively. with

components and

subscript cs

s

denote

the steady mean

is the local sound speed in the flow.

The flow The

problem is then to solve (4.1) subject to

u(0,t) = f(t)

(4.2)

B(q,u) (l,t) = 0

(4.3)

Acoustic Source

•.

-I

0

Termination

•

_

••

N.I N+2

I

I

I

I

I

0

0.25

0.5

0.75

1.0

Axial Position,x/L

Figure 2

-13-

where in (4.2), f(t) (4.3)

B

denotes a source which varies in time.

In

is a linear operator which will give no reflections at

the exit (i.e., at x = i). As mentioned in the introduction (4.3) is derived upon linearization of (4.1). This process yields

(q)t = 0

+A

U

(4.4)

X

where

A

(us

=

Cs2/q s

The eigenvalues of this matrix are

u

s

+ c

s

and

u

s

- c

s"

The first one is positive and the other is negative.

These signs

give the characteristic directions of propagation.

We form the

matrix

T

from the eigenvectors so that

T-I A T is diagonal.

Then the characteristic variables are

(vl) =

v2

This gives

T-I

.

(4.5)

-14-

_ Vl

q + u qs cq (4.6)

_

v2 where

vI

u

Cs

corresponds to the positive eigenvalue and

negative one. which

q +

qs

is

set

At the right boundary to

zero

to

obtain

vI

v2

to the

is the inflow variable

the

nonreflective

boundary

condition (4.3), i.e.,

B(q,u) (l,t) -

We

remark

here

discussed

that

for

the

q qs

u _ 0. Cs

fully

two-dimensional

in the next section a similar procedure

problem

is used to

obtain this type of boundary condition. As far as the source term is concerned it is driven with a harmonic input f(t) = A cos t

where

A

is the amplitude of the source.

Once again the numerical procedure described in Section II is applied

to get accurate smooth solutions.

For high Mach

numbers and high source strengths the artificial viscosity which was discussed in that section was added to obtain solutions. As mentioned in the introduction the goal in this class of prob/ems is to study the attenuation of sound pressure level at the exit

section.

It was experimentally observed

that

high

-15-

source

strengths

reasons. and

and

This

is

a discussion Using

(lower boundary

V.

is

ongoing reported

are

recently

in

one

situation We

the

equations

given

mean

and

last

here

of

Let

corresponding

mean

The

Their

scheme

treatment

of

ours.

pressure

The

a

u, v,

field

a

has

also

Then

of

of

our

will

be

combination

procedure

been is

of

studied

similar

configuration

equations

equations flow p

then p

velocity 's'

on

the

is determined

we are

and

and

component

flow.

part

procedure

using

physical

field

subscript

is

solution

last

of

to this

3.

perturbed

y

the

the

effects

This

solution

These

Let

in

two-dimensional

solutions

Our

the

the

described

situations

in Figure with

work

of

section.

velocity,

respectively.

solution

numerical

14.

propagation.

numerical 12.

state.

component

the

seek

wave

situation.

Euler

in

Similar

is depicted begin

possible

numerical

different

from

are

VI.

reference

details

reference

nonlinear

(5.1).

duct

elsewhere.

in

in

to

Complete

in

this

flow

Model

us

theory

in

with

different

motivated

asymptotic

mean

in Section

available

obtained

work.

the

demonstrated

results

nonlinearities

of

equations

Two-Dimensional

section

the

Euler

conditions

The

the

being

number

is available

full

order)

Mach

the

use

to

derived

density

the

from

acoustic and

quantities

from

from

subtracted

denote

field

simulate

equations isentropic

a x

pressure

denote are

the given

relation

-16-

as given in (5.2).

B-_ %-_ Psu + UsP + pu + _

_-_ psu + UsP + pu

+ -_

+ -_

psv + VsP + pv = 0

2 0sU2 )

2PsUsU + Usp +

+ 2UsPu + p

PsVsu + PsUsv + UsVsP + Psuv + UsPV - VsPU

= 0

(5.1)

_-_ psv + VsP + pv

+ @--_PsUsv + PsVsu + UsVsP

+ PsUV + UsPV + VsPU )

+ _-y @ (2psv sv + Vsp 2 + psV2 + 2v spv + p) = 0

P = c2[ Is

where

+ /_

c2 = yps/Ps

(P/Ps)IP

(5.2)

is the local sound speed in the flow.

The

system (5.1) has the form @8

@F

@G

@--_ + _-_+ @--_ = 0.

where

_

=

(81,

82 , 83 ) and

(5.3)

-17-

81 = P 82 = psu + UsP + pu B3 = psv + VsP + pv

u,v and

p

are determined by

U =

82 - us 81 Ps + B1

(5.4) V =

83 - vs 81 Ps + B1

and P =

81 •

YJ

Inflow _ Boundary_

y= d(x)

AcousticWaves

Inlet Wall /

i....--Termination I Boundary

IL._

._L!

MeanFlow _--x

I

II I

Figure 3

Two-Dimensional

-18-

The area variation (see Figure 3) is included as follows.

Let

the contour of the area variation be given by y = d(x).

We introduce new variables

=

X

n= so that n = 1

y

,

(5.5)

will be the surface of the inlet.

This change of

variables allows us to do the computation in a rectangle, but the system (5.3) takes the form

-_

(dS)

+ -_

(dF) + -_

(G - _Fd')

= 0.

(5.6)

This system is solved together with an inflow condition and a nonreflective

condition

at

exit

plane.

On

the axis

the

y

component of the velocity is set to zero and on the wall normal component of the velocity is zero.

To derive the nonreflective

condition at the exit plane we considered the variation of (5.6) only in the

x

direction, i.e.,

(d_8)+ _

(dF) = 0.

(5.7)

-19-

This

equation

variables

as

condition

that

is

then

indicated comes

the

means

the

in

out

•

For

linearized

of

to

Section

this

obtain

IV.

procedure

characteristic

The

nonreflective

is

82 - (cs + us)81 = 0.

source

plane

pressure

wave

incident

is prescribed.

(5.8)

conditions

Upon

are

linearization

used of

which

(5.2)

we

have _ p(O ,y,t) P

C

2 S

or

81 = p(0,y,t) 2 c

(5.9)

s

This completes numerical sample

VI.

the

statement

of

the

problem.

Again

the

scheme described in Section II is applied to get a

solution

reported

in the

next

section.

Discussion of Results For the sound radiation problems discussed in Section III

the computations were performed typical grid sizes were at

eight diameters

origin. distance

on a Cyber-203 machine.

115 × 35.

(diameter of

The

The source boundary was kept the cylinder)

away from

the

The sound pressure levels were calculated on a circle at 10 diameters away

from the open end.

The farfield

-20-

boundary was chosen to enclose this circle. results with

a

result of

Wiener-Hopf technique.

reference 12 which

was

done

using

An experimental study to simulate this

situation is available in reference 15. spinning mode

We compared our

In this experiment a

synthesizer was used to produce both plane and

spinning mode waves.

A sample result comparing our results with

both results of references 12 and 15 is presented in Figure 4.

0 -5 -I0( -15 -20 -25 dB, Level -30

0 [] 0

-35 -40 -45 -50 -55 -60 -65 0

I I0

I 20

ExperimentalData(ref 15) SavkarTheory(ref 13) Numerical ka = 3.370 Mode= 2

I I I I I I 30 40 50 60 70 80 Angle,deg

I 90

Figure 4

This comparison was made for a wave number the spinning mode number

m = 2.

ka = 3.37

and for

In this result theoretical, ex-

perimental and numerical results are in good agreement, except

-21-

near

the

fact

that

other

origin

for

in the and

to

reference

For

the

compare

an were

interested

in

were

with

of

the

to Mach

comparisons

the

throat

gas

explicitly.

numbers

Solid the x/L

-.90 lines

are

asymptotic = 1.00

For

refer

for the

this

higher

one

on

a

computations reference

(i -

in

IMtl),

is shown dB

source

results

of

our

results

respectively

at

the

where

L

This

examples

the

steady us

a

is

the

length

.75

of

the

and

throat

a throat

=

the

asymptotic

frequency

x/L

be

typical

levels

The

at

ohecan

the

the

5 for

and

with

pressure

computations.

is the

linearly

-.50

Mt

the

contour

the

Since

where

station

of

was

theory

with

The

particular

compared

I0.

in Figure

140

a

Ps and

are

sound

this

I0.

for

by

are

comparisons

ii.

configuration

The we

accelerates

satisfied

at

reasons

to

number

to

made.

description

flow

Mach

attempt

reference

reference

entry

in

the

In particular

equations

arises

of

applied

mean

an

8) was

A detailed

the

Euler

model

of

was

in

the

A comparison and

IV

throat.

developed

chosen

number.

we

However,

theory

that

approaching

was

is one

duct.

here

The

situation

control

comparisons

model.

available

a way

dynamic

nonlinear

number

is

(Mt).

theory

more

(reference

This

Section

at the

asymptotic

Mach

in

given

-.90

dimensional

good.

asymptotic

duct

one

to the

to completely

nonlinear

result

Crocco-Tsien

number

parameter

For

two-dimensional

in such

and

Mach

so

the

the

is designed

solved

not

called

contour

waves.

is due

4.

discussed

geometry

This

it is difficult

experimental

comparisons

procedure

plane

results.

quasi-one-dimensional

with

made

experimental

experiment

modes

readers

the

Mach

452

Hz.

others

are

and duct.

at

-22-

-8

[]

Asymptotictheory (ref 10), x/L = 0.75

O

Asymptotictheory(ref 10), x/L = 1.00 Finite Difference

-4 u

(l_lMtl_Z o %4

-.8

I

I

7T/2

_T

I

3zr/2

I

27T

Time,t

Figure 5

Shock results are also in good agreement (see reference 6).

As

far

is

as

the

sound

reduction

at

the

termination

section

concerned a result for a higher sound pressure level (156 dB) for the source

is presented

in Figure 6.

In this case acoustic

shocks occur and cause energy loss and we can see at 5 dB sound pressure level drop at the termination section.

Mt = O.90 180 Sound 170 Pressure Level,dB 160

150 0

0.25

0.50 Axial Distance,x/L

Figure 6

0.75

I 1.00

-23-

Around this source pressure level (156 dB) Figures 7 and 8 for absolute value of throat Mach numbers increasing from .70 to 90 we have shown that the pressure level reduction is increasing and the

wave

forms

starting

from a

smooth stage

becomes steeper showing the shock phenomena.

and

ultimately

Figure 7 shows

acoustic suppression and Figure 8 shows the distortion of wave form.

This

validates

the

experimental

suggestion

that

the

increase in the Mach number attenuates sound at the termination or the exit section.

180 -

170 -

MachNo.(Throat) .70 .80 .90 .85

//_

SPL 160

I50

140 0

I 0.2.5

I I 0.50 0.75 Axial Position, xlL

I 1.00

Figure 7

[

-24-

3.0 -

MachNo.(Throat) 30 .80 .85 .90

1.5 \

AcousticVelocity 0

7 ;t /

-.

\ !

_.k__..l__-

..--;--j;

L,

..

/

-1.5

-3.0

I M2

0

I _" Time,t

--// I 3;T/2

I 27r

Figure 8

For the two-dimensional model we discussed in Section V we needed

a

two-dimensional

flow.

To

simulate

a

situation we

considered again the one-dimensional flow solution us and Ps but we introduced the y component of the velocity according to

d'(x) Vs(X,y) = _

y •

This is valid (see reference 12) provided the variation of is small.

(6.1)

d(x)

For the two-dimensional case the shocks were predicted

at low Mach numbers (reference 12).

Here we present a sample

result that for a throat Mach number -0.85 and a 135 dB source. Here we observed a i0 dB sound pressure with reduction on the

-25-

axis

at

the

reduction

was

termination observed

Langley

(reference

will

reported

be

8). in our

section.

in an As

The

experimental we

mentioned

forthcoming

paper.

same study

before

pressure conducted the

full

level at NASA results

-26-

References

[i] Astley, R. J. and W. Eversman, "Wave Envelope in Infinite Element Schemes for Noise Radiation from Turbofan Inlet," AIAA-83-0709 (1983).

[2]

Bayliss, A. and E. Turkel, "Radiation Boundary Condition for Wave-Like Equations," Comm. Pure AppI. Math., 33, No. 6 (1980), pp. 707-725.

[3] Gottlieb, D. and E. Turkel, "Dissipative Two-Four Methods for Time-Dependent Problems," Math. Comp., 30 (1976), pp. 703-723.

[4] Hariharan, S. I. and A. Bayliss, "Radiation of Sound from Unflanged Cylindrical Ducts," ICASE Report No. 83-32, NASA CR-17217 (1983), to appear in SIAM J. Sci. Statis. Comput., 1984.

[5]

Hariharan, S. I. and H. C. Lester, "A Finite Difference Solution for the Propagation of Sound in Near Sonic Flows," J. Acoust. Soc. Am., 75 (4) (1984).

[6] Hariharan, S. I. and H. C. Lester, "Acoustic Shocks in a Variable

Area

Duct

Containing

Near

Report No. 83-64, NASA CR-172274 Comput. Phys., 1984.

Sonic Flows,"

ICASE

(1983), to appear in J.

-27-

[7]

Horowitz, Finite

S. J., Element

Radiation 0124

[8]

Jones,

M.

G.,

R.

C.

Inlets

in

Zinn,

for

"An

Iterative

Predicting

Steady

3

(1980),

Flight,"

Sound AIAA-82-

K.,

a Nearly

Myers,

M.

K. of

Sound

Myers,

M.

K. of

S. D.

Acoustic

(1975).

Sound

Inlet,"

M.S.

(1982).

"A

Problems,"

Finite

Element

Internat.

Flow,"

A.

J.

J.

and

51

K.

in

Method

Math.

Sound

AIAA-81-2012

Calleghari,

Acoustic

Vib.,

and

Marin,

Choked

Math.

(4),

Transmitted

(1981).

"On

the

Theory

in

Near

(1977),

pp.

517-531.

Uenishi,

Through

a Near

Singular

Sonic

"Two-Dimensional

Transmission

AIAA-84-0497

Savkar,

P.

Development

Sound

Number

University

"Shock

Linear

J.

Mach

of

311-350.

and

Flows,"

Investigation

Subsonic

S.

pp.

Through

Flow",

High

Interface

M.

Analysis

Experimental

and

Myers,

Duct

B. T.

Technique

Washington

Exterior

Behaviour

[13]

a

George

Sci.,

[12]

"An

in

MacCamy, for

[ii]

Turbofan

and

(1982).

Thesis,

[i0]

Sigman

Integral

from

Attenuation

[9]

R. K.

Duct

Nonlinear Sonic

Throat

(1984).

I. H.

Modes

Edelfelt,

with

Flow

"Radiation Mismatch,"

of Cylindrical NASA

CR-132613

-28-

[14]

[15]

Walkington

N.

J.

and

Solutions

to Shocked

Ville,

M.

of

J.

the

Bellmouth

and

Radiation Including

W.

Eversman,

"Finite

Waves,"

AIAA-83-0671,

Acoustic

R. J. of

Silcox,

Sound

the

Flow

from

"Experimental an

Effect,"

(1983).

Investigation

Unflanged NASA

Difference

Duct

TP-1697,

and 1980.

a

I.ReportNo. NASA

CR-172434

ICASE Report

2. Government AccessionNo.

3. Recipient's Cat_logNo.

No. 84-39

4. Title and Subtitle

5, Report Date

Numerical Solutions of Acoustic Wave Problems Using Euler Computations

Propagation

August 1984 6. Performing Organization Code

7. Authoris)

8. Performing Organlzation

S. I. Hariharan

Report No.

84-39 10. Work Unit No.

NameandAddress for Computer Applications

9. Performing Organization

Institute

in Science

and Engineering

11. Contract or Grant No.

Mail Stop 132C, NASA Langley Hampton, VA 23665

Research

Center

NASI-17070 13. Type of Reportand PeriodCovered Contractor Report

12. Sponsoring Agency Name and Address

14. Sponsoring AgencyCode 505-31-83-01

National Aeronautics and Space Administration Washington, D.C. 20546 15. Supplementary Notes

Langley Technical Final Report

Monitor:

R. H. Tolson

16. Abstract

This paper reports solution procedures for problems arising from the study of engine inlet wave propagation. The first problem is the study of sound waves radiated from cylindrical inlets. The second one is a quasi-one-dimensional problem to study the effect of nonlinearities and the third one is the study of nonlinearities in two dimensions. In all three problems Euler computations are done with a fourth-order explicit scheme. For the first problem results are shown in agreement with experimental data and for the second problem comparisons are made with an existing asymptotic theory. The third problem is part of an ongoing work and preliminary results are presented for this case.

r

J17. Key Words (Sugg_ted by Authoris}}

18. Distribution

acoustics, inlets, finite differences, Euler computations, shock waves

64 71

Statement

Numerical Acoustics

Unclassified

19. Security_a_if. (ofthisreport) Unclassified

20. Security Cla_if.(of this_ge) Unclassified

Analysis

- Unlimited

21. No. of Pages 30

22. Dice A03

ForsalebytheNational Technical Information Service, Springfield, Virginia2216t

NASA-Langley, 1984