j-

NASA Technical Memorandum 109118

/

Evaluation of Several Non-reflectingComputational Boundary Conditions forDuct Acoustics

Willie R. Watson and William E. Zorumzki

Langley Research Center, Hampton, Virginia

Steve L. Hodge

Virginia Consortium of Engineering and Science Universities, Hampton, Virginia

(NASA-TM-I09118) EVALUATION OF

SEVFRAL NON-REFLECTING

COMPUTATIONAL BOUNDARY CONDITIONS

FOR DUCT ACOUSTICS (NASA. LangleyResearch Center) Z6 p

83/71

N94-33685

Unclas

0012105

May 1994

National Aeronautics and

Space AdministrationLangley Research CenterHampton, Virginia 23681-0001

https://ntrs.nasa.gov/search.jsp?R=19940029179 2020-05-23T11:41:30+00:00Z

Evaluation of Several Non-reflecting

Computational Boundary Conditions for Duct Acoustics

Willie R. Watson and William E. Zoru,uski

NASA I,an!lh'y ltesrazrh ('r,lcr. IhrmldOn. ViT_linia 23666

Stew' I,. I lodge

Virqinia Consortium of E,!liueerin 9 and Science llnim,rsilies, llampton Virginia 2.q666

Abstract

Several non-reflecting computational Imundary conditions that meet certain criteria and

haw, potential applications 1,o du<'t acoustics are evaluated for their effectiveness. The same

interior solution scheme, grid, and order of apl)roximation are used to evaluate each condition.

Sparse matrix solution techniques are applied t.o solw' the ,watrix equation resulting from

the discretization. Modal series solutions flu' the sound attenuation in an infinite duct are

used to evaluate the accuracy of each non-reflect lug boundary condition. The evaluations are

performed for sound propagation in a softwall duct, for sew'ral sources, sound frequencies,

and duct lengths. It is shown that a recently developed noulocal boundary condition leads

to sound attenuation predictions considerably more accurate l,han the local ones considered.

Results also show that. this condition is more accurate for short ducts. This leads to a

substantial reduction in the nllnlher of grid points when coml)ared to other non-reflecting

conditions.

1 Introduction

Ow'r the past decade, there has l>een consid,.,'al_h, interest in the vse of computational

methods for obtailfiug solutions to i_rohh.ms in a,'roa<'oustics. This interest stems primarily

from a lack of exact analytical solutions for i>re_licting, umlerstandiug, and controlling various

acoustic phenonlena. As conqmtatioual ,m,d,'ls have ew)lw'd, dilficulty with the closure of

the compl,tational domain has etlte,'_ed as a tl|a.jor I>rohlem in the calculations.

The difficulty with closure of the COml,,latio]ml domain arises I>ecause aeroacoustics

problems are typically set in an infinite clonmi,i with a radiation condition on the boundary

surface at infinity, lh>wever the computatio,m.l domain cannot extend to infinity, so the

In'ohlem is decoml)osed into a liuit.e COmln,tatioual domain within some o,,ter domain that

extends to the boundary al. infinity. The interface between these domains is called the

artificial or computational boundary. One needs a set, of equations valid at this computational

boundary. Tile primary purposes of these equations is to guarantee, a unique and well-see

posed solution to the aeroacoustics problem. These equations are usually a set of partial

differential operators whose terms involve only local information at each boundary point

such as tile dependent variables and their derivatiw,s. 'lb avoid nonphysical reflections from

occurring at the computational boundary, these differe.ntial operators are also intended to

constrain the local solution to consist of waves traveling outward from the computational

domain. When used in this manner, tl,e differential eql,ations are called local non-reflecting

bot,ndary conditions.

There currently exist a large nmuber of research papexs concerned with the development

of local non-reflecting conditions for use at, computational boundaries (refs. [1]- [6]). The

references cited above represent only a small sampli,lg of local boundary conditions, which

have potential application to duct acoustics. I lnfi)rtunately, experiences show that the con-

ditions developed in these works are reflecting fi)r a large class of aerocoustic problems.

This is especially true for classes of a_','oacoustic problems fi)r which waves impinging on the

computational boundary is not close to normal incidence.

The proper apl)roach to dealing with the i)roblem of finite computational domains is

to match the computatio,|al soll,tiol, it, the in,mr domain with tile general solution in the

outer domain. This g('neral ol,ter solul, io|l satislies I,he radial, ion condition at infinity and

constrains the solution on the COml)utational interface. But this constraint is not in general

a local condition. Instead, a given value of a variable at one point on the interface surface

influences the values of other variables at all points on the interface surface. Constraints of

this type are call nonlocal conditions. They can be const,'ucted for all classes of problems in

which the exterior domain is linear.

In a recent report (ref. [7]), the authors l)resented the formulat;ion of a nonlocal non-

reflecting boundary conditio,i fi,r duct acoustics. The purpose of this paper is to compare

this nonlocal boundary conditio*l to s,,w,ral local con(lil, io,ls that coul(i compete with it. Due

I,o space limitatio,i a, ld I,imc ,'estraints, all of tl,c existing non-rell('cti||g boundary conditions

cm,ld not be coral)areal wil.h tl,is ,lew co,lditio|l. 3'o linfit the number of choices, we include

only those conditions that meet certain criteria.

Section 2 defines the I)asic ('qual.ions used i,i the computation and presents the non-

retlecting boundary conditions that are teste(I. Modal series solutions for tile sound atten-

nation in an infinite (l_,ct are used to crab,ate the accuracy of each boundary condition.

TI,is series solution is also presented in section 2. The interior solution technique, method

of implementing each boundary condition, elfect of each condition on tile matrix structure,

and matrix solution technique is described in section 3. Results for a broad range of acoustic

parameters are presented in section 4. Conclusions, relative to boundary conditions evalu-

2

ated in this paperaregiven in section5. Refi,rencesand figurescited in this report aregivenat the end of section5.

2 Governing Equations and Boundary Conditions

Consider a two-dimensional rectangular duct without mean flow as shown in figure 1. The

duct is assumed infinitely long ill tile axial direction with a known acoustic source pressure

at the plane x = 0. Tile walls of the duct contain sound absorbing material whose material

properties vary along the axis of the duct for 0 < :r < L. Sound absorbing properties of the

wall lining are specified by perscribing the admittance of the lower wall, Be(z), and upper

wall,/_,,t(x), of tile duct. Within the region L < .r < co the material properties of the liner

are assumed uniform, so that an outgoing wave fiehl exists in this region. It is the purpose

of this work to test several non-reflecting bou,ldary conditions for te.rminating the duct at

.T-_-- Z.

Steady-State acoustic waw, solutions within the duct in figure 1, take the form

_(._,y, t)= ,(:_,.v),Y_'I' (1)

where lfi is the steady state acoustic pressure, f is the frequency in llertz, i = _/_-], and

p(z, y) satisfies llelmholtz's equation

V'zp + k'zp = 0 (2)

llere, k = _ is the freespace waw'm,mbe,', c is the sound speed and _72 is the Laplacec

operator in the (z, y) plane.

The inflow and wall bot,nda, ry conditions require a specification of the source pressure

and wall admittance

p(0,:,_)= c;(u) (3)

e_P(:,,,o) = -,:_:/_o(:,')p(:r,o) (4)

()P , ,,

_t.. tt) : ,:_¢_,,(:_)1,(:_.n) (5)

where G(y),/_0(z), and fln(:r) de.note the k,lown source pressure and normalized admittance

of the lower and upl)er wall respectiw,ly. To comphfl,e the. specification of the boundary value

problem in the duct, a non-reflecting condition illust be specilied at the outflow boundary.

Several non-reflecting boundary conditions were considered for application at x = L. In

order to limit the number of choices only those boundary conditions that met the following

criteria were considered

3

1. The boundary condition must haw_'a f,'equencyand possibly time domain extension.

2. The boundary condition must be extendible to sl,earing flows and three dimensionality.

3. The order of the boundary co,Mition nmst be such that the linear finite element theory

of ref. [7] could be applied for the interior solution.

4. The coupling of the I)oundary condition with the interior solution scheme must lead

to a coefficient matrix that remains I)lock tridigonai.

The non-reflecting conditions that m(s.t the above criteria that are tested in this paper are

!. The local boundary condition of Giles (ref. [5])

1 ol) i)_c/)--t+ _ = 0 (6)

2. The noniocal freq,wncy-do,,,ai,_ i,o,,,,da,'y ('o,,,lition of tl,e current authors fief. [7])

{i,,)= (7)

where {p_}, {uj} are vectors ,'ontaining values of the frequency-domain acoustic pres-

sure and normal veh)city at, boun(lary node i arm j respe('tively. Further, the coefficients

in the nodal impedan(',' matrix [Zo] are defined exl,licitly in ref. [7].

3. The local highly-ahsorlfing bo, m(lary co,Mition of Engquist and Majda (ref. [1])

÷ -o (8)t.2 ?)Z2 c i);ri)l 2 ?)y2

Several rernarks concerni,,g the alJov,, bo,mdary conditions are in order. Each of the above

bo.ndary conditions has be, m specialized to both a ,,o-flow environment and right moving

waves. All of the above co,.liti(ms are ,.m-,','lhwti,,g for plane wave sources. The first

no,,-reflecting condition above is a familiar one that Wa._ (lerived by lledstrom (ref. [2]) and

sew'ral others prior to the work of (lih.s. This condition ha._ been used by duct acousticians

for sew,ral decades as a terminal, iota condition, a,M is often referred to as the pc termination

condition. Because the Giles condition (ref. [5]) in the absence of mean flow reduces to

this condition, it is roferred to here as the "C, ih's condition." Finally, Engquist and Majda

((ref. [!]) have shown that the third condition giw_s reflections considerably smaller than the

first condition when the Neumann wall I)oundary condition is imposed. This conclusion may

not be valid when tl,e sound propagates between walls lined with sound absorbing material.

4

Periodic acoustic fields determined fi'om the non-rellecting boundary conditions above are

used to evaluate the performance of tile wall lining over x length of linings. The following

expression is used to evaluate the acoustic int,,,n,sity at a point (x,y) in the duct ref. ([8])

( r'"">]"I;(x,y)- i _ .(_r,?/)[,. _ ) O)4rfpo

wl,ere the superscript asterisk denotes the complex conjugate, p0 is the mean density within

the duct, and _{ } denotes the real part of the complex expression within the braces. The

attenuation over x length of li,ling in decibels is

dH(x)= lOt,,:/,ow(°)W(x) (10)

fo IiW(x) = l(x,y)dy (11)

Modal series solutions for the attenuation are used to (letermine the accuracy of each

non-reflecting boundary condition. These series solutions are possible when the material

properties of the wall lining are constant. When this condition is met, the solutions to

Hehnhoitz's equation in the fi)rm of outgoing waves are of the complex exponential form

_(x,y) = _--_M A,,,p,,,(y)eih.,,,: (12)m=l

where K,, is the axial propagation constant, and the functions Pro(y) are the acoustic pres-

sure eigenfilnctions. Note thai, the series has I)een truncated at a finite number M. To

insure no reflections, the sum in equation (12) is taken only over modes whose axial prop-

agation constants possess l)ositive ifJ_aginary parts. The method for ol)taining Pm and h'm

is described in ref. [7] and tim mode aml)litu&, coel[icients are ol)tained from the source

condition and the orthogonality of tit(, li,wcl (tl,ct modes

A,,,= fd__:(:,/)I',,,(:/),#/.Ii_'"" _,,a, (13)" m _ ,, ) .']

Equations (12) and (13) are substituted into equations (9)-(11) to obtain the modal series

expression for the attenuation of I,hc lining.

3 The Numerical Method

In this section of the i)al)er we d_'scril)(, l,he inl,('rior solution t('chniqoe, the numerical im-

ph'mentation of each of l.he non- reflecting boundary conditions, and the matrix solution

te(,h,fique. Several details l,aw' been pl,rposely omitted since they can be obtained in several

of the cited references.

3.1 The Interior Solution Technique

Tile numerical method chosen to solve equation 2 coupled with the source, wall and non-

reflecting boundary conditions is a Galerkin Finite l,lement" Method with linear elements

used as the basis functions. The method is descril)ed in detail in the earlier paper (ref. [7]).

Application of tile method results in a matrix equation of the form

[A]{_} = {F} (14)

where [A] is an MNxMN complex matrix, and {_} is a MNxl column vector containing

the nodal values of the unknown acoustic pressure. The integers N and M are the number

of grid points in the x and y direction respectively. Equation (14) does not contain the

effects of the non-reflecting condition. The non-reflecting condition must be imposed on this

matrix equation before the solution can be obtained. Details of the implementation of each

non-reflecting boundary condition are now discussed.

3.2 Boundary Condition Implementation

The frequency-domain form of the Giles condition ref. [5] is

t: I,(L, u) - i

The Giles condition is discretized a.s followed

_I,( L, y)-0 (15)

Oz

k PN,j -- i [lINd -- PN-I,.i]= O, j = 1,2...M (16)

(3rN -- J'N_i)

Note that the spatial gradient iq the boundary ('ondition discretization is only first order

accurate. Thus, the interior solution and I)o, ndary condition discretization are of the same

order. Equation 16 constitutes M eql,ations which are imposed as restraints on equation 14

prior to solving the matrix equation.

Boundary condition imph'mentation of I,he nonlocal condition is exactly as discussed in

rcf. [7]. The axial w;Iocity w','tor at th,,. grid points {uj} is expressed in terms of the gradient

of the acoustic pressure fiehl. The axial acoostic pressure, gradient is then discretized with a

lit'st order difference approximatio,I. This I'esllll,s in M restraint equations which are imposed

on the matrix equation in the usual manner.

The frequency-domain hwm of the l,ngqmst and Majda condition ref. [1] is

k._l,(L,y) _ ikOP(L,y) 10"_p(L,y)+ -0 (17)i)x 2 Oy 2

6

For tile numerical experiment presented here, the above equation is discretized using first

order one-side differences

k2pN.j -- ik [PN,j -- PN-I,j]

(xN xN-,)Jr [PN,j+2 - 2pN,.i+, Jr PN,j] = O,

2(yj+, -- yj)2j = 1,2...M- 2 (lS)

k2pN,j _ ik[PN,j -- PN-I,j] [PN,j-2 -- 2pN,j-] + PN,j]

(Z N __XN_I) Jr 2(yj --y__l) 2 =0, j = M -1,M (19)

Note that although tile boundary condition has a second derivative term in y, first order one-

sided differences are still used to approxinlat,e all derivatives in x and y. The M equations

generated by tile (liscretizatio, al)ow.d are iml)ose(I o. the matrix equation (14) a.s a set of

restraints.

3.3 Effects of Boundary conditions on the Matrix Structure

The augmented global matrix generated by Galerkin's Method, following application of the

source, wall, and non-reflecting boundary conditions, is an unsymmetric, positive indefinite,

complex matrix. Fortt, nately, owing to the discretization scheme anti choice of non-reflecting

boundary conditions, it will be block tridiagonal as shown in figure 2. The superscript T

denotes the matrix transpose in the figure. Each minor block At and Bt are MxM matrices

that are tridiagonal. These blocks are obtained from the interior solution scheme and wall

boundary conditions. The mi,or block Et is the identity matrix, which results from applying

the source condition (3). Mi,or blocks CN anti DN are MxM complex matrices that contain

the effects of tile non-refle('ting con(litions. I;_a(,h minor block CN and DN is a diagonal

matrix when the Giles condition (16) is i,lq)lemented. Application of the Engquist and

Maj(la condition (18) and (19) lea(Is to a diagonal minor block CN, while DN will contain a

main diagonal, two super(liagonal, and two sul)diagonals. Finally, both minor blocks are full

matrices when the nonh)cai co,(litio, of Zor,nlski, Watson, and llo(Ige (7) is implemented.

3.4 Solution to the Matrix Equation

The matrix [A], gevwrate(! I)y the n.'l, ho(I (h'sc,'i bed h(,re, is not symmetric or positive definite.

Fortunately, it is block tridiagonal as shown in ligure 2. Much practical importance arises

from this structnre, a.,_ il, is (,onw,nient for minimizing storage and maximizing computational

elliciency. Economy of storage is a.cheived I)y storing the rectangular array of coefficients

within the bandwidth of [A] as sl,ow,i i, ligl,re 3. All computation, storage and boundary

condition implementation is i)erformed o,ly on the rectangular portion of tiffs matrix. Special

matrix techniques exist fi)r a sol,tio, of this structure. Gaussian elimination with partial

pivoting and equivalent row inlinity norm scaling is ,sed to re(lute the rectangular system to

upper triangular form. Back substitution is then ,,s('d to obtain tile solution for the acoustic

pressure.

4 Results and Discussion

A computer code implementing all three boundary conditions and tim modal series solution

discussed in this paper has been developed and programmed to run on a supercomputer. The

results were obtained with the underlying objective of comparing the attenuation obtained

from each boundary condition with the modal series solutions over a broad range of acoustic

parameters. The same interior solution schenlc, grid, and order of approximation are used

to evahlate e_h non-reflecting condition. With|,1 this context, the effect, of changing the

source pressure, frequency and duct length is (waluatcd fi)r a specified lining.

With respect to the acoustic parameters, effects of three source pressures on the boundary

condition are studied (see eq. 12)

(:(y) = v,(v)

G(v) = v (v)

c,(v) = )lrl| _ I

Other parameters include a softwall dn<:t with (rio = /5:t

(20)

(21)

(22)

= .5- .5i), three frequencies

(f = 100 Hz, f = 1,000 Ilz, f = 5,000 liz) and fot,r duct lengths (L = H, L = .8H, L =

.611, L = .211). All calculations were i)(.rfi)rm(,(I for a duct 1 meter higll (H = 1 rn) and at

ambient conditions (c = 343 _'r/SCC). The g,'iding remained fixed as each acoustic parameter

was changed. Fifty-one evenly sl)a('cd points wer(' us(.'d in the y direction (M = 51) and

tit(, number of axial points N was detormincd such that 10 points per axial wavelength were

retained at the highest freq,e,lcy of 5,{)0{} llz.

Figure 4 compares the attonuations p,'cdic'tcd from the boundary conditions when the

location of tl,e bounda,'y is at, L = II. The source is the lowest order mode and the frequency

is 100 llz. The new no,llocal condition giw' predictions that are in excellent agreement with

tl,e modal series. Gih's's condition is mor,, accurate tl,an the condition of Engquist and

Maj(la for _ < .65, and in th(, region _ > .65 the condition of Engquist and Majda is

more accurate. All curw.s eXCel)t that of l",ngq,ist and Majda show a linear attenuation

rate, a.u expected for a si,lgh, |node source. The oscillations in the attenuations predicted

by the Engquist and Majda condition are typical of results obtained when the boundary

is reflecting. Figures 5,6 and 7 show results for the same source and frequency, but when

the boundary is located at x = .811, x = .6I!, and x = .2H, respectively. Although the

attemmtions are lower due to the sl,orter duct length, the trends are consistent with those

8

of tigure 4. Note also that, as the boundary is brought closer to the source, the attenuation

on a percentage basis is less accurate for all conditions except the nonlocal condition.

Attenuation predictions in figure 8, for which L - H and the frequency is 1,000 Hz

are typical of results obtained for sew+ral other duct lengths at that frequency, llere, the

source is still the lowest order mode. At this higher frequency the liner is not effective, giving

little attenuation over the lining. Both the (tiles and the nonlocal condition gives accurate

predictions but the condition of Engquist anti Majda gives poor comparison with the modal

series. Results were also computed for 5,000 llz with the lowest order mode as the source

and for several duct lengths. The attenuation curves are not shown in order to limit the

amount of discussion, ltowever, trends were consistent with that of figure 8.

Figure 9 shows results at a frequency of 100 llz, L = H, and for the fifth order mode

source. All boundary conditions give good predictions for x/L < .75. llowever, the condition

of Engquist and Majda and also that of ('.iles gives poor attenuation predictions near the

outflow boundary. In contrast, the nonlocal conditio, is in excellent agreement with the

modal series in this region. Tim discrepancy i, the attenuation predictions obtained with the

other two boundary conditions was not elimi,ated by applying the boundary conditions closer

to the source. To the contrary, the discrepancy on a percentage basis increased when the

boundary was moved closer to the source, i"urther, predictions with the nonlocal condition

were equally accurate when the bou,dary condition application point was moved close to

the source.

Figure l0 shows the predictions at 1,000 llz for the same source and duct length as

figure 9. The boundary condition of Eugquist and Majda gave poor predictions and this curve

is not shown. The two remaining boundary conditions give accurate predictions, although

the nonlocal condition is closer to the modal series results. Figure 11 shows predictions

when the frequency ix i,,-r,,a,s,,,I to 5,000 Iiz an,t all boundary condition curves included.

Note that the liner performs poorly at, this frequency giving little attenuation. All boundary

conditions give good 1)redictions at l,his frequency. Note that the curve for the nonlocal

condition and the modal series solul, ion are identical.

it should be noted that in [7], res,lts at a frequency of 5,000 Hz could not be accurately

I)redicted using the no.local I)ounda.'y comlitio.. Further, it was speculated that the poor

prediction at this freq.ency was a result of .sing only 3.4 points per axial wavelength. The

banded solver adopted in this paper allows several hundred thousand degrees of freedom to

I)e incorporated with relative, ease. The good agreement with the modal series predictions

at 5,000 llz using the no.local boundary condition confirms that this conjecture was true.

Turbomachinery sources, such as aircraft engine fans, are distributed sources. Such

sources contain acoustic energy in many (!uct modes. Figure 12 compares attenuation pre-

/.1

¢' 9

JORI._INAL PAGE IS

OF POOR QtJALil'Y

dictions for the distributed source define by (22) at a frequency of 100 Hz and for L = H.

The nonlocal condition is generally the most accurate for this source. Further, the condition

of Engquist and Majda gives attenuation predictions closer to the modal series results than

the condition of Giles. Figure 13 shows results for tl,e same source when the frequency is

increased to 1,000 Hz. The curve generated by the Engquist and Majda condition is not

shown, since it led to predictions ten decibels higher than the modal series results. Both

the Giles and the nonlocal condition give nearly the same predictions, although the Giles

condition is slightly more accurate in a small region near the end of the duct.

5 Concluding Remarks

Several non-reflecting boundary conditions which have l>otential application to duct acoustics

have been evaluated for their effectiwmess. Those that met the criteria and were tested are

1. The local boundary condition of Giles.

2. The local highly-absorbing i)oundary condition of Engquist and Majda.

3. The nonlocal I)oundary condition of Zorumski, Watson and lh)dge.

The case of of an infinite two-dimensional +lnct witho,t flow was used for simplicity. All

thr<_, boundary conditions however, have extensions to thr_ dimensionality, variable area

and wall linings, and flow. The interior solution tech,fique was a linear finite element method.

All three boundary conditions were tested using the same grid and order of approximation.

A band solver has been used to account for the large number of degrees of freedom required

fo,' high frequency and hmg ducts.

The effectiveness of each bou,da,'y condition has been evaluated by comparing predicted

attenuation in a softwall d_,ct with analytic_d results available from modal theory. All three

boundary conditiol,s were evaluated h)r the lowest order mode, a higher order mode, and for

a distributed source. Atten,atio, I>redictions for several frequencies were evaluated, and the

effects of applying each boundary condition close to the source has been investigated.

Results presented here show that the new ,o, lo<:ai boundary condition of Zorumski,

Watson, and llo<lge gave+ results consistent with exact analytical predictions over a broad

range of acoustic parameters. This I_on,l<lary condition gave attem,ation predictions more

ac<'urate than the <:on<litio, of Giles and ,,ore accurate than the condition of Engquist and

Madja over a range+ of acoustic paranlel.ers. Giles condition however, was competitive at

higher frequencies where the, li,ler was not effecl, ive. The boundary condition of Engquist

and Majda gave poor predictions h>r the range+ of parameters considered. The accuracy of

10

tile attenuation predictionson a percentage basis were observed to decrease with duct length

when the Giles or condition of Engquist and Majda were used. tlowever, the accuracy of the

attenuation predicted using tile nonlocal boundary condition is accurate for short ducts as

well. This is an important result, since a substantial reduction in grid points can be obtained

by applying the nonlocal condition close to the s(n,nd source. Implementation of the band

solver has confirmed that the poor attenuation predictions at 5,000 llz in the earlier paper

is a result of having too few points per wavehmgth in the discretization.

6 References

References

[1] Engquist,Bjorn anti Majda, Andrew., "AI)sorl)ing Boundary Conditions for the Numer-

ical Simulation of Waves," Mathematics of Computations, Vol. 31, p. 629-651, 1977.

[2] lledstrom, G.W., "Nonreflecting lloundary Conditions for Nonlinear tlyperbolic Sys-

tems," Journal of Compuhttio.aal Physics, Vol. 30, !). 222-237, 1979.

[3] Thompson, Kelvin W., "Time Dependent Bountlary Conditions for tlyperbolic Sys-

tems," Journal of Computalional Physics, Vol. 68, p. !-28, 1987.

[4] Bayless, A. and TurkeI,E., "Radiation Boundary Conditions for Acoustic and Elastic

Wave Cah:ulations," Communications on Pure and Applied Mathematics, Vol. 32, p.

312, 1979.

[5] Giles, M.B., "Nonrefle('ting llo, n(lary Conditions for Euler Equation Calculations,"

AIAA Paper 89- 1912, CP, 1989.

[t;]Watson, Willie IL and Myers, M.K., "Time I)el)entlent Inflow-Outflow Boundary Con-

ditions for 2-I) Acoustic Systems," AIAA .Journal, Vol. 2q, no. 9, Sept. 1991 p. 1383-

1389.

[7] Zorumski, W.E, Watson, W.R., a,ld lhMg_' S. I_., "A Non-reflecting Boundary Conditions

for Duct Acoustics ," NASA TM 109091, April, 1994.

[8] Eversman, W., "l",nergy How (',riteria for Acot,stic 1)ropagation in Ducts with Flow,"

Journal of the Acouslicai Socit_iy of America, Vol. 49, No. 6, June 1971.

II

H

Source Pressure,

r;'(y)

Upper wall admittance, [_n(x)

//////,9////!

Non-reflectingBoundary __..

\\\\\\\\\\"q

Outgoing waves

V / / ///// 7 // h,\\\\\\\\\'qLower wall admittance, _So(z)

a L •

x

Figure 1: Two dimensional duct and coordinate system.

12

MN

El

I

B3

A3 ]TJ3

A4 ]TJ4

I]TN_3 AN-2 BN-2

AN-1 BN-1

CN DN

MN

Figure 2: Structure of the global stifFn¢,ss matrix, [A], with minor blocks.

13

........---3 M + 1 ___.

0 I_'_ 0

"Y

//,T A4 /34

• • •

• • •

• • • MN

• • •

• • •

• • •

I__:_ AN-._ BN-,_

i_I-2 AN-I 111N-1

CN DN 0

Figure 3: 'File storc(I M Nx(3M+l)re('tangular cocmcient matrix with nonzero coefficients.

14

4

3

00.0

Modal Series

• Giles

................. Engquist & Majda

..................... Zorumski, Watson, & Hodg,

0.2 0.4 0.6 0.8 1.0

Dimensionless axial coordinate, x/L

Figurc, 4: Attenuatiovl ('Oml)ariso,i fi)r I,h_, Iow_,nt order mode" source in a softwall duct at 100

//: (1,=II).

15

4

I" Modal Series/

I" ......... Giles i

!" ................. Engquist & Majda >

3 •

/ ../:--jjs S_

s o_"

p s • o,_,s °•"

e'l ...."• so• •

S •° s

j so S •

joJ°

So g°

0

0.0 0.2 0.4 0.6 0.8 1.0

Dimensionless axial coordinate, x/L

Figure 5: Attenu_tiol= comparison for t,h_, lowest ord_'r mode source in a softwall duct at 100

llz (I, = .SH).

16

3

"O

u_

"10C--

mm

¢-O

(D

4-

Modal Series

......... Giles

................. Engquist & Majda

..................... Zorumski, Watson, & Hodge

00.0 0.2 0.4 0.6 0.8 1.0

Dimensionless axial coordinate, x/L

Figure 6: Attenuation comparison for the Iow_,st order mode source in a softwall duct at 100

Hz ( L = .611).

17

4

O0"10

u_

3

e-

¢1)

0

Modal Series

......... Giles

................. Engquist & Majda

..................... Zorumski, Watson. & Hodge

/a

/s

-- l/

!!

./I

di

i

Jl"

I"i 0

,i

0.0 0.2 0.4 0.6 0.8 1.0

Dimensionless axial coordinate, x/L

Fi.a;ure 7: Attenuation compari._on for t,lm low_,st order mode so_,rce in a softwall duct at 100

ii,,, (/, = .21I).

18

4

3

cG

"_2 -"10¢.-

8

==1Q)

Modal Series

......... Giles

................. Engquist & Majda

i ..................... Zorumski, Watson. & Hodge

!

!

!!!!!!

I

i!I

i0

I

iiit!I!!!I

iiIi

t

/!

!!

!/

s

s* o

fs °

S'

s •

s •

_.o 'S

,_•_"

s s'_"

• ° °_'_s_ ,S

j°P,_'' I I l I I .... I .... I .... I0

0.0 0.2 0.4 0.6 0.8 1.0

Dimensionless axial coordinate, x/L

Figure 8: Attenuation compariso,i to=' the lowest order mode source ill a softwall duct at

1,000 llz (L = II).

19

,oo _ r.....-tI- Modal Series l

=" ......... Giles //.. .....

::.-..:::::.::.-..::.z_;0.u,:,,_wT;°,_ +/"

401"-" /

0 , i i , I , , , i I , , , , ! I I i i I

0.0 0.2 0.4 0.6 0.8 1.0

Dimensionless axial coordinate, x/L

I°'ignre 9: Attennat, ion conilia.rison for the [ifl, h order nio(le sonrc(; in a softwall duct at 100Hz (/, =//).

2O

cO"10

c_

e--o

8

6

4

2

00.0

Modal Series......... Giles

.....................Zorumsk_, Watson,

0.2 0.4 0.6 0.8 1.0

Dimensionless axial coordinate, x/L

I"i/qlr,. I0: Al,l,,.,,,al, io,, ,'ol,i)ari,_o. fi,r l.h,, lifl, h .rd,'r ,,i,,d(, so,,'c(, iu a soft.wall d,,rt, at 1,000//._ (/, =/!).

21

o.s[- /

f i,e °j °S°S.J JModal Series ./"_

0.4 V .Giles ./"/

r ................. Engquist & Majda //¢

CID r ..................... zorumski, watson, & Hodge.//

$ o..b ../

.=_o0._I- ,/== _ ,,2"

/ ,._

I: ,/

0ol ",,__,, i

O0 , i i i I , , , , I , , , , I , , , , I

0.0 0.2 0.4 0.6 0.8 1.0

Dimensionless axial coordinate, x/L

Figure 11: Attenuation comparison for the Iifl, h order inode source in a softwall duct at 5,000

l/z (L = il).

22

30

"10

cff

t--t--O

CD

10

Modal Series......... Giles

................. Engquist & Majda.....................Zorumski, Watson, & Hodge

00.0 0.2 0.4 0.6 0.8 1.0

Dimensionless axial coordinate, x/L

Figure 12: Attenuation COml_arison for a distributed source ill a soft, wall duct at 100 Hz

(L = .).

23

12

10

El

E:up

86

:=3

2

00.0

Modal Series......... Giles

.....................Zorumski, Watson,

& Hodge

0.2 0.4 0.6 0.8 1.0Dimensionless axial coordinate, x/L

Figure 13: Atl,enuat, ion comparison for a disl, rilmt,ed source in a softwall duct at 1,000 Hz

(/, = n).

2,t

REPORT DOCUMENTATION PAGEForm A_provedOMB No. 0704-0188

Publlo repotting burden lot tkll 0olieelion ol Infomv_lo_ is ullmated to average 1 hou_ per reqx_m, lm:iuding Ihe tlmo I_ revk_ Inelructlonl. Imlmhlng exkding data oo_oes.olll_llfl_ _ rl'JlllttlllNn 0 t_e d*Jl fteO_l)d, ir1_ _ _ rOvieW_1_ |hi O01kK_IIOR Ol Inlurmallo_t. Send 0¢m_1 fel_cllTt 0 Ihls burden eSlllr_ or" arty other llipecl oi th_l

oolkdm ¢_ Inlorm4lllon, Inoludlh0 I_)g_Uonl for tod_if_ INs burden. Io WlhlnWon Headquanene Se_vloei, Dlte¢lo_io lot"Inlormltlon OpefldlOml im¢l Reports, 1215 Jeffwlmn DavisH_hway, 8u_ 1_)4, Adlnglon, VA 22202-4302, Ind to the O(floo o_ Minigemord and Budget, P_10enIKJ<k Roducllon ProJl¢l (0704-01B), Wl_llngton, DC 20603.

1, AGENCY USE ONLY (Lelve blank) 2. REPORT DATE 3, REPORT TYPE AND DATES COVERED

May 1994 Technical Memorandum4. TITLE AND IMJBTITLE S. FUNDING NUMBERS

Evaluation of Several Non-reflecting Computational Boundary Conditions 538-03-12-01for Duct Acoustics

L AUTHOR(S)

Willie R Watson, William E. Zorumski, and Steve L. Hodge

7. PERFORMINGORGANIZATIONN/LME(S)ANDADORESS(ES)

NASA Langley Research CenterHampton, VA 23681-0001

L SPONSORINGI MONITORINGAGENCYNAME(S)ANDADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

B. PERFORMING ORGANIZATION

REPORT NUMBER

10. SPONSORING/MONITORINGAGENCY REPORT NUMBER

NASA TM-109118

11. SUPPLEMENTARY NOTES

Watson and Zorumski: Langley Research Center, Hampton, VAHodge: Virginia Consortium of Engineering and Science Universities, Hampton, VA

12L DISTRIBUTION/AVAILABILITYSTATEMENT

Unclassified - Unlimited

Subject Category 71

12b. DISTRIBUTION CODE

13. ABSTRACT (MaJlmurn2OOwordl)

Several non-reflecting computational boundary conditions that meet certain criteria and have potentialapplications 1o duct acoustics are evaluated for their effectiveness. The same interior solution scheme, grid, and

order of approximation are used to evaluate each condition. Sparse matrix solution techniques are applied tosolve the matrix equation resulting from the discretization. Modal series solutions for the sound attenuation in an

infinite duct are used to evaluate the accuracy of each nonreflecting boundary conditions. The evaluations areperformed for sound propagation in a softwall duct, for several sources, sound frequencies, and duct lengths, it

is shown that a recently developed nonlocal boundary condition leads to sound attenuation predictions

considerably more accurate for short ducts. This leads to a substantial reduction in the number of grid points

when compared to other nonreflecting conditions.

14. SUBJECT TERMS

Duct Liners; Computational boundary conditions; Non-reflecting boundaryconditions; Finite element method; Helmhollz equation; Sparse marix technique

17. SECURITY CLASSIFICATION

OF REPORT

Unclassified

;111. SECURITY CLASSIFICATION

OF THiS PAGE

Unclassified

19. SECURITY CLASSIFICATION

OF ABSTRACT

NSN 7640-01-280-5500

15. NUMBER OF PAGES

25

16. PRICE CODE

A03

20. LIMITATION OF ABSTRACT

StandMd Form 298 (RAY. 2-89)Pmoodbed by ANSI SKI. Z30-18298-102