AIAA JOURNA L
Vol. 33, No. 10, October 1995
Computational Aeroacoustics: Issues and Methods
Christophe r K. W. Tarn *
Florida State University, Tallahassee, Florida 323063027
Computationa l flui d dynamic s (CFD) ha s made tremendou s progres s especiall y in aerodynamic s an d aircraf t
desig n ove r the past 20 years. An obviou s questio n to as k is "why no t us e CF D method s to solv e aeroacoustic s
problems?" Mos t aerodynamic s problem s ar e time independent, wherea s aeroacoustic s problem s are, by definition,
time dependent. Th e nature, characteristics, an d objective s of aeroacoustic s problem s ar e also quit e differen t fro m
the commonl y encountere d CF D problems. Ther e ar e computationa l issue s tha t ar e uniqu e to aeroacoustics.
For thes e reason s computationa l aeroacoustic s require s somewha t independen t thinkin g an d development. Th e
objective s of thi s pape r ar e twofold. First, issue s pertinen t to aeroacoustic s tha t ma y or ma y no t be relevan t to
computationa l aerodynamic s ar e discussed. Th e secon d objectiv e is to revie w computationa l method s develope d
recentl y tha t ar e designe d especiall y fo r computationa l aeroacoustic s applications. Som e of th e computationa l
method s to be reviewe d ar e quit e differen t fro m traditiona l CF D methods. The y shoul d be of interes t to the CF D
an d flui d dynamic s communities.
Nomenclature
#0 = spee d of soun d
D = jet diamete r at nozzl e exi t
er = uni t vecto r in the r directio n
ee = uni t vecto r in the 9 directio n
/ = frequenc y
L = cor e lengt h of a jet
p = pressur e
u = velocit y componen t
U j = jet exi t velocit y
a = wave numbe r
a = wav e numbe r of a finit e differenc e scheme
P = wav e numbe r in the y directio n
A £ = time step
AJ C = size of spatia l mes h
8 = thicknes s of mixin g laye r
X = acousti c wav e lengt h
va = artificia l kinemati c viscosit y
p = densit y
co = angula r frequenc y
a ) = angula r frequenc y of a finit e differenc e scheme
o)i = imaginar y par t of the angula r frequenc y
I. Introduction
I
T is no exaggeratio n to say tha t computationa l flui d dynamic s
(CFD) has mad e impressiv e progres s durin g the last 20 years,
especiall y in aerodynamic s computation. In the hand s of competen t
engineers, CFD has become not onl y an indispensabl e metho d for
aircraf t load predictio n but also a reliabl e desig n tool. It is incon 
ceivabl e tha t futur e aircraf t woul d be designe d withou t CFD.
Needles s to say, CFD method s hav e been ver y successfu l for
the clas s of problem s for whic h they wer e invented. An obviou s
questio n to ask is "why not use CFD method s to solve aeroacous 
tics problems?" To answe r thi s question, one mus t recogniz e tha t
the nature, characteristics, and objective s of aeroacoustic s prob 
lems ar e distinctl y differen t from thos e commonl y encountere d in
aerodynamics. Aerodynamic s problem s are, generally, time inde 
pendent, wherea s aeroacoustic s problem s are, by definition, time
Receive d Nov. 1, 1994; presente d a s Pape r 95067 7 at th e AIA A 33r d
Aerospac e Science s Meeting, Reno, NV, Jan. 912,1995; revisio n receive d
Marc h 13, 1995; accepte d fo r publicatio n Marc h 15, 1995. Copyrigh t ©
199 5 b y Christophe r K. W. Tarn. Publishe d b y th e America n Institut e o f
Aeronautic s an d Astronautics, Inc., wit h permission.
* Professor, Departmen t o f Mathematics. Associat e Fello w AIAA.
dependent. In mos t aircraf t nois e problems, the frequencie s are ver y
high. Becaus e of these reasons, ther e are computationa l issue s tha t
are relevan t and uniqu e to aeroacoustics. To resolve thes e issues,
computationa l aeroacoustic s (CAA ) require s independen t thinkin g
and development.
An importan t poin t needs to be made at thi s stage. Computationa l
aeroacoustic s is not computationa l method s alone. If so, it woul d
be calle d computationa l mathematics. The applicatio n of compu 
tationa l method s to aeroacoustic s problem s for the purpos e of un
derstandin g the physic s of nois e generatio n and propagation, or for
communit y nois e predictio n and aircraf t certification, is the most
importan t par t of CAA. The proble m area may be in jet noise, air 
fram e noise, fan and turbomachiner y noise, propelle r and helicopte r
noise, duc t acoustics, interio r noise, soni c boom, or othe r subfield s
of aeroacoustic s (see Ref. 1 for details). Computationa l method s are
the tool s but not the ends of CAA. It is aeroacoustic s tha t define s
the area.
As yet ther e has not been widesprea d use of computationa l meth 
ods for solvin g aeroacoustic s problems. Thi s paper, therefore, con
centrate s on discussin g the methodolog y issue s in CAA in the hope
of stimulatin g interes t in CAA application s and furthe r develop 
ment s or improvement s of computationa l methods.
The firs t objectiv e of thi s pape r is to discus s issue s pertinen t to
aeroacoustic s tha t ma y or ma y not be relevan t to computationa l
aerodynamics. To provid e a concret e illustratio n of thes e issues,
the case of direc t numerica l simulatio n of supersoni c jet flow s and
nois e radiatio n wil l be used. The secon d objectiv e is to revie w re
centl y develope d computationa l method s designe d especiall y for
CAA applications.
Befor e one design s a computationa l algorith m for simulatin g
supersoni c jet nois e generatio n and radiation, it is importan t tha t
one has some idea of the physic s of supersoni c jet noise. Thi s is
extremel y important, for any computationa l scheme woul d have a
finit e resolution. Thi s limitatio n prevent s it fro m bein g capabl e of
resolvin g phenomen a associate d wit h fine r scale s of the problem.
The principa l component s of supersoni c jet nois e are the turbu 
lent mixin g noise, the broadban d shockassociate d noise, and the
screec h tones.2'3 In a supersoni c jet, the turbulenc e in the jet flo w
can be divide d int o the largescal e turbulenc e structures/instabilit y
wave s and the finescal e turbulence. Bot h the large turbulenc e struc 
ture s and the finescal e turbulenc e are nois e sources. However, it is
known 23 tha t for hot jets of Mac h numbe r 1.5 or highe r the large
turbulenc e structures/instabilit y wave s are responsibl e for the gen
eratio n of the dominan t par t of al l of the thre e principa l component s
of supersoni c jet noise. In the discussio n tha t follows, it wil l be
assume d that the nois e fro m finescal e turbulence, bein g less impor 
tant, is ignored. The resolutio n of finescal e turbulenc e is, therefore,
not a primar y issue.
1788
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1789
II. Issues Relevant to CAA
To illustrat e the various computationa l issues relevan t to CAA,
we wil l consider the case of direct numerica l simulatio n of the gen
eration and radiatio n of supersoni c jet noise by the large turbulenc e
structures/instabilit y wave s of the jet flow. Since a computatio n do
mai n must be finite, it appear s that a good choice is to select a
domai n nearl y identica l to tha t in a physica l experiment. Figur e 1
shows a schemati c diagram of a supersoni c jet noise experimen t in
side an anechoic chamber. The exit diameter of the nozzle is a natura l
length scale of the problem. In order that microphon e measurement s
do provide representativ e farfiel d noise data, the lateral wal l of the
anechoi c chambe r shoul d be placed not less tha n 40 diameter s fro m
the jet axis. For a highspeed supersoni c jet, the centerline jet veloc
ity in the full y developed region of the jet decays fairl y slowly, i.e.,
inversel y proportiona l to the downstrea m distance. Thus, even at a
downstrea m distance of 50 jet diameters, the jet velocit y woul d still
be in the moderatel y subsoni c Mac h numbe r range. To avoi d stron g
outflo w velocit y and to contai n all of the noiseproducin g region
of the jet inside the anechoi c chamber, it is preferabl e to have the
wal l wher e the diffuse r is located to be at least 60 diameter s down 
stream fro m the nozzle exit. The preceding consideration s defin e
the minimu m size of the anechoi c chambe r that wil l be used as the
computationa l domain.
A. Large Spectral Bandwidth
Jet noise is broadband, and the spectru m is fairl y wide. Figure 2
shows a typica l noise spectrum of an imperfectl y expande d
entrainmen t flo w
X
Fig. 1 Schematic diagram showing a supersonic jet noise experiment
conducted inside a ventilated anechoic chamber. Not shown are the
soundabsorbing wedges on the walls of the chamber. The anechoic
chamber is the ideal physical domain for direct numerical simulation.
130
11 0
90
70
50
screec h ton e
0.03
0.1 1
Strouhal number, St = fD/Uj
Fig. 2 Typical farfiel d noise spectrum of an imperfectly expanded su
personic jet, measured at 30deg inlet angle, showing the three principal
noise components. Data from Seiner.4 Nozzle design Mach number 2.0.
Jet Mach number 1.5.
supersoni c jet measure d by Seiner.4 The discret e componen t at the
center of the spectru m is the screech tone. The peak to the right of
the screech tone is the broadban d shockassociate d noise. The low
frequenc y peak to the lef t is the turbulen t mixin g noise. The screech
tone and the broadban d shockassociate d noise exist onl y for im
perfectl y expande d jets when a quasiperiodi c shock cell structur e is
present in the jet plume. For perfectl y expanded jets, the noise con
sist s of turbulen t mixin g noise alone. Generally, the Strouha l numbe r
range (Strouha l numbe r = fD/Uj) of interest spans fro m 0.01 to
10 or a ratio of 103 between the highest and the lowest frequency.
The spatia l resolution requiremen t of the jet (region A of Fig. 1) is
dictated by the sound wave s wit h the shortes t wavelength s or the
highes t frequencies. Typicall y a minimu m of six to eight mesh point s
per wavelengt h is required. Suppos e one is interested in the turbu 
lent mixin g noise of perfectl y expande d supersoni c jets alone. In thi s
case, the maximu m Strouha l numbe r of interes t may be taken as 1.0,
givin g an acousti c wave lengt h A approximatel y equal to the jet di 
ameter D. A simpl e calculatio n will show that even by using a spatia l
resolutio n of onl y six to eight mesh point s per wavelength, the num
ber of mesh point s inside the computatio n domai n is enormous. How
to develop finit e differenc e algorithms that will give adequat e resolu
tion at six to eight mesh point s per wavelengt h is an importan t issue.
B. Acoustic Wave/Mean Flow Disparity
The rootmeansquar e velocit y fluctuatio n associate d wit h the ra
diated sound is usuall y quite small compared wit h that of the mean
flo w of the jet. For example, for a Mach 1.5 jet the measure d sound
intensit y at 40 jet diameter s awa y is around 124 dB. This gives a rati o
of sound particl e velocit y to the jet exit velocit y of about 1.5 x 10~4.
That is, the velocit y fluctuation s of the radiated sound are fou r or
ders of magnitud e smaller than the mean flow. This large disparit y
between acousti c and flo w variable s present s a severe challenge to
direc t numerica l simulation. The smal l magnitud e of the acousti c
disturbance s can, perhaps, be better appreciate d by notin g tha t it
is usuall y smaller than the error (differenc e between the compute d
mean flo w and the exact mean flo w solution) incurre d in the com
putatio n of the mean flow. This observatio n led Roe5 to state tha t
"there is a fea r amon g investigator s that the acousti c solution s may
be hopelessl y corrupte d by computationa l noise." This issue raise s
the questio n of whethe r it is mor e pruden t to solve for the perturba 
tions afte r the mean flo w has firs t been determined or to solve the ful l
nonlinea r equation s to captur e the ver y smallamplitud e soun d fiel d
directly. For the jet noise problem, especiall y for screech tones, the
nonlinearit y of the proble m is crucia l to the nois e generatio n pro
cess. Thus, ther e is no alternativ e but to fac e the largemagnitud e
disparit y issue.
C. Distinc t and WellSeparate d Length Scales
Jet nois e simulation is an archetypica l multilengt h scal e aeroa 
coustics problem. In the noise source region, the growt h and decay
of the large turbulenc e structures/instabilit y wave s are controlled
locall y by the thicknes s 8 of the mixin g layer. However, becaus e the
flo w spreads out in the downstrea m direction, they are influence d
globall y by the core lengt h L of the jet. Outside the flow, the natura l
lengt h scale of the acousti c fiel d is the wave lengt h A. For most su
personi c jets, these variou s lengt h scales are ver y distinc t and well
separated, typically we have 8 <^ X < L. The existenc e of ver y
disparat e lengt h scales calls for carefu l consideratio n of the spatia l
resolutio n requiremen t befor e a direct numerica l simulatio n is at
tempted. Near the nozzl e exit, region B of Fig. 1, the halfwidt h of
the mixin g laye r thicknes s is usuall y foun d to be abou t 5% of the jet
diameter. To resolve adequatel y the instabilit y wave s in the mixin g
layer of the jet, a minimu m of say 15 mesh point s are needed. This
gives Ar = 0.0033D wher e Ar is the radial mesh spacing in the
mixing layer. In the acousti c field, region A of Fig. 1, if sound wave s
of Strouha l numbe r 1.0 are considered, the mes h spacin g require d
is Ar = D/6. Thus a spatia l resolution of roughl y 50 times fine r is
needed in the sound source region than in the acousti c field. Since
numerica l instabilit y of most finit e differenc e schemes occur s whe n
f he CF L numbe r is large r tha n a critica l value, it follow s tha t the
computatio n time step is dictated by the size of the fines t mesh. This,
needlessly, leads to excessive CPU time. To make CAA practical,
1790
TAM
methods that would overcome the curse of disparat e length scales
are ver y much needed.
D. Long Propagation Distance
The quantitie s of interest in aeroacoustics problems, invariably,
are the directivit y and spectrum of the radiated sound in the far
field. Thus the compute d solutio n mus t be accurat e throughou t the
entir e computatio n domain. This is in shar p contras t to aerodynam 
ics problems wher e the primar y interes t is in determinin g the load
ing and moment s acting on an airfoi l or aerodynami c body. In this
class of problems, a solution that is accurat e only in the vicinit y
aroun d the airfoi l or body woul d be sufficient. The solutio n does
not need to be uniforml y accurat e throughou t the entir e compu 
tatio n domain.
The distance fro m the noise source to the boundar y of the com
putatio n domai n is usuall y quite long. To ensure that the compute d
solution is uniforml y accurate over such long propagatio n distance,
the numerica l scheme mus t be almos t fre e of numerica l dispersion,
dissipation, and anisotropy. If a large numbe r of mes h point s per
wavelengt h ar e used, thi s is not difficul t to accomplish. However,
if one is restricted to the use of onl y six to eight mesh point s per
wavelength, the issue is nontrivial. To see the severit y of the re
quirement, let us perfor m the followin g estimat e for the jet nois e
problem. Numerica l dispersio n error is the resul t of the differenc e
betwee n the group velocitie s (not the phas e velocit y as commonl y
believed) of the waves associated wit h differen t wave number s of
the finit e differenc e equations and that of the original partial dif 
ferentia l equations. Assume that the computatio n boundar y is at
40 jet diameter s away. Let a (a) be the wave numbe r of the finit e
differenc e scheme (see Sec. Ill or Ref. 6 for the definitio n of a).
Then the grou p velocit y of the acousti c wave s of the numerica l
scheme is give n by (da/da)a 0 (assumin g the numerica l scheme
is dispersio n relatio n preserving). The time needed for the soun d
wav e to propagat e to the boundar y of the computatio n domai n
is 40D/00  Thus, the displace d distanc e due to numerica l disper 
sion is [(da/da)ao — ao](40D/a0). If a mesh of six spacing s per
jet diamete r is used and an accumulate d numerica l displacemen t
less than one mesh spacing is desired, then the slope of the a (a)
curve of the numerica l scheme must satisf y the stringen t require 
ment of
da
I
40x6
(1)
Most loworde r finit e differenc e schemes do not satisf y the preced
ing condition.
Finite differenc e schemes, invariably, have builti n numerica l dis
sipatio n arisin g fro m time discretization. This cause s a degradatio n
of the compute d sound amplitude. Suppos e AdB is the acceptabl e
numerica l error in decibels. Then, it is eas y to show tha t if &>/ is
the imaginar y part of the angula r frequenc y of the numerica l time
marchin g scheme, thi s conditio n can be expressed mathematically 6
as
240 '
(2)
In the case of AdB = 1.0 and the CourantFriedrichsLewy numbe r
Atao/Aj c = 0.25, it is straightforwar d to fin d co t &t > —1.2 x
10~4. Ver y few timemarchin g scheme s can meet thi s demandin g
requirement.
£. Radiation and Outflow Boundary Conditions
A computatio n domai n is inevitabl y finit e in size. Becaus e of this,
radiatio n and outflo w boundar y condition s are require d at its artifi 
cial boundaries. These boundar y condition s allow the acousti c and
flo w disturbance s to leave the computatio n domai n wit h minima l
reflection. Agai n let us conside r the proble m of direc t numerica l
simulatio n of jet noise radiatio n fro m a supersoni c jet as show n in
Fig. 1. The jet entrains a significan t amoun t of ambien t flui d so that
unless the computatio n domai n is ver y large, there will be nonuni 
for m timeindependen t inflo w at its boundaries. At the same time,
the jet flo w must leave the computatio n domai n throug h some part
of its boundary. Along thi s par t of the boundary, there is a steady
outflow. It is well known that the Euler equation s suppor t three types
of smallamplitud e disturbances. They are the acoustic, the vortic
ity, and entrop y waves. Locally, the acousti c waves propagat e at a
velocit y equal to the vector sum of the acousti c speed and the mean
flo w velocity. The vorticit y and entrop y waves, on the other hand,
are convecte d downstrea m at the same speed and directio n as the
mea n flow. Thus, radiatio n boundar y condition s are require d alon g
boundarie s wit h inflo w to allow the acousti c wave s to propagat e out
of the computatio n domai n as in region C of Fig. 1. Along boundarie s
wit h outflo w such as region D of Fig. 1, a set of outflo w boundar y
condition s is required to facilitat e the exit of the acoustic, vorticity,
and entrop y disturbances.
F. Nonlinearities
Most aeroacoustic s problems are linear. The supersoni c jet noise
problem is an exception. It is known experimentall y whe n the jet
is imperfectl y expanded, stron g screech tone s ar e emitte d by the
jet. The intensit y of screech tones aroun d the jet can be as high as
160 dB. At thi s high intensity, nonlinea r distortio n of the acousti c
wavefor m is expected. However, because of the threedimensiona l
spreadin g of the wave front, experimenta l measurement s inside ane
choi c chamber s do not indicat e the formatio n of shocks. Thus, in
the acousti c field, a shockcapturin g scheme is not strictl y required.
Althoug h there are no acousti c shocks, inside the plume of an im
perfectl y expande d jet, shocks and expansio n fan s are formed. These
shocks are known to be responsibl e for the generation of screech
tones and broadban d shock noise.2'3 These shocks are highl y un
steady. The use of a good shockcapturin g scheme tha t does not
generat e spuriou s numerica l wave s by itsel f is, therefore, highl y
recommende d in any direct numerica l simulatio n of noise fro m
shockcontainin g jets.
G. Wal l Boundary Conditions
The impositio n of wall boundar y condition s are necessar y when 
ever there are solid surface s present in a flo w or sound field. Accurat e
wal l condition s are especiall y importan t for interior problems such
as duct acoustics and noise fro m turbomachinery. For the super
sonic jet noise problem, solid wal l boundar y condition s are needed
to simulat e the presence of the nozzl e as shown in Fig. 1.
It is easy to see, unless all of the firstorde r spatia l derivative s
of the Euler equation s ar e approximate d by firstorde r finit e differ 
ences, the order of the resulting finit e differenc e equation s woul d be
highe r tha n the origina l partia l differentia l equations. Wit h highe r
order governin g equations, the numbe r of boundar y condition s re
quired for a unique solution is larger. In other words, by using a
highorde r finit e differenc e scheme, an extende d set of wal l bound 
ary condition s mus t be developed. The set of physica l boundar y con
ditions, appropriat e for the origina l partia l differentia l equations, is
no longe r sufficient. Asid e fro m the need for extraneou s boundar y
conditions, the use of highorde r equation s implies the generatio n of
spuriou s numerica l solution s near wal l boundaries. In the literature,
the questio n of wal l boundar y condition s for highorde r schemes ap
pears to have been overlooked. The challenge here is to fin d way s to
minimiz e the contaminatio n of the unwante d numerica l solution s
generated at the wal l boundaries.
III. Computation of Linear Waves
Recently, a numbe r of finit e differenc e schemes6"9 has been pro
posed for the computatio n of linea r waves. Numerica l experiment s
and analytica l result s indicat e tha t onl y highorde r scheme s are ca
pable of calculatin g linear waves wit h a spatial resolution of six to
eight mesh point s per wavelength. The highorde r essentiall y non
oscillator y (ENO)10 and the dispersionrelationpreservin g (DRP)6
scheme s are two suc h algorithms. The ENO scheme is wel l known.
Here we wil l discus s the DRP scheme and in doing so introduc e a few
concept s tha t are new to CFD. The DRP scheme was designed so tha t
the dispersio n relatio n of the finit e differenc e scheme is (formally )
the same as tha t of the origina l partia l differentia l equations. Accord
ing to wave propagatio n theory,11 this would ensure that the wave
speeds and wav e characteristic s of the finit e differenc e equation s
are the same as those of the origina l partia l differentia l equations.
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1791
A. Wave Number of a Finite Difference Scheme
Suppose a sevenpoint central differenc e is used to approximat e
the firs t derivative 3f/dx at the ah node of a grid wit h spacing
AJC; i.e.,
(3)
Equation (3) is a special case of the followin g finit e differenc e equa
tion wit h j c as a continuou s variable:
j = 3
The Fourier transfor m of Eq. (4) is
3
(4)
(5)
where ~ denotes the Fourier transform. By comparin g the two sides
of Eq. (5), it is evident that the quantit y
(6)
is effectivel y the wave number of the finit e differenc e scheme Eq. (4)
or Eq. (3). Tarn and Webb6 suggested to choose coefficient s aj so
that Eq. (3) is accurat e to order (A*)4 whe n expande d in Taylor
series. The remainin g unknow n coefficien t is chosen so that a: is a
close approximatio n of a over a wide band of wav e numbers. This
can be done by minimizin g the integrate d erro r
E =
\ctAx — aAx\2d(aAx)
(7)
Tar n and Shen12 recommende d to set 7 7 = 1.1. The numerica l value s
of cij determine d this way are given in the Appendi x togethe r wit h the
coefficient s for backwar d differenc e stencils. Backwar d differenc e
stencils are needed at the boundaries of the computation domain.
Figure 3 shows the relation a AJ C vs a AJC. Over the range a A* up
to 1.0 the curve is nearl y the same as the straight line a = a. Figure 4
shows the slope da /da as a functio n of a Ax. Clearl y da/da is close
to 1.0 (withi n 0.3%) up to ot AJ C — 0.9 (or seven mesh point s pe r
wavelength). This satisfie s the requiremen t of Eq. (1). The standar d
0.0 0.2
0.6 0.8
aAx
Fig. 4 da/da vs a AJ C for the DRP scheme, ——; and the sixthorder
standard central difference s cheme,..........
sixthorde r scheme can resolve waves up to a Ax = 0.6 (10.5 mesh
point s per wavelength). There is, therefore, an obvious advantag e
in using the DRP scheme.
B. Angular Frequency of a Finite Difference Scheme
For time discretization, Ref. 6 proposed to use the followin g fou r
levels marchin g scheme:
(nj)
(8)
wher e the superscrip t indicate s the time level. The Laplace transfor m
of Eq. (8) wit h zero initial condition (for nonzero initial condition,
see Ref. 6) yields
1)
dt
(9)
wher e ~ represent s Laplac e transform. Th e Laplace transfor m of the
time derivative, i.e., the right side of Eq. (9), is equal to —icof. On
comparin g the two sides of Eq. (9), the quantit y
1)
(10)
1.2 1.6 2.0
aAx
Fig. 3 OL AJ C vs a AJ C relation for the standard central difference second
order,.........; fourth order,     ; sixth order, — • —; and the DRP
scheme, ——.
is identifie d as the effectiv e angula r frequenc y of the time marchin g
scheme (8). The coefficient s bj are determined by requiring Eq. (8)
to be second order accurate. Tar n and Webb6 foun d the remainin g
coefficien t by minimizin g a weighte d integra l error, whic h force s a)
to be a good approximatio n of CD. The numerica l values of bj ar e
given in the Appendix.
For a given value of cbAt, Eq. (10) yields fou r roots of CD At. In
order that the scheme is numericall y stable, all of the root s mus t
hav e a negativ e imaginar y part. Numerica l investigation s revea l that
this is true as long as cbAt is less than 0.4. Hence by choosing a
sufficientl y smal l At, the scheme is stable. A detailed discussion of
the numerical stabilit y of the DRP scheme is provided in Ref. 6. The
numerica l dissipatio n rate of the finit e differenc e scheme is given
by the imaginar y par t of a>. By means of Eq. (10) it is, therefore,
possible to estimate a priori, for a particular choice of the time step
At, the amoun t of numerica l dampin g that would occur (see Ref. 6).
This informatio n is most valuable in the design of computer codes.
C. Group Velocity and Numerical Dispersion
The DRP scheme was formulate d so that the form s of the dis
persio n relation s are preserve d in the discretizatio n process. For the
linearized Euler equations, the dispersion relations for the acoustic
wave s in two dimension s in the absence of a mean flo w are
a) = ±a ()(ot 2 +
(11)
1792
TAM
The corresponding dispersion relations for the DRP scheme are
[obtaine d by replacing co, a, and ft by &>, a, and ft in Eq. (11)]
The group velocity 11 of the acousti c wave s of the DRP scheme can
be obtaine d by differentiatin g Eq. (12) wit h respect to a and ft. It is
straightforwar d to fin d
da '
±a()
da 
da '
If a smal l At is used in the computation, the n cb ~ a; so tha t
(do)/d&> ) ~ 1.0. For plan e acousti c wave s propagatin g in the x
direction (ft = Q), the wave velocity given by Eq. (13) reduces to
3 CD da
— = ±007
3 a da
(14)
It is clear fro m Eq. (14) and Fig. 4 tha t differen t wav e number s wil l
propagat e at differen t speeds. The dispersivenes s of a numerica l
scheme is, therefore, dependen t largel y on the slope of the numer 
ical wave number curve. For the sevenpoin t DRP scheme, da/da
deviate s increasingl y fro m 1.0 for a Ax > 1.0 (see Fig. 4). The wav e
speed of the shor t wave s (hig h wave number ) is not equal to #0 
In fact, for the ultrashor t waves (a Ax ~ TT ) wit h wavelength s of
about two mesh spacing s (gridtogri d oscillations ) the group veloc
ity is negativ e and highl y supersonic. The shor t wave s are spuriou s
numerica l waves. Once excited they woul d contaminat e and degrade
the numerica l solution.
To illustrat e the effec t of numerica l dispersion, let us consider the
solutio n of the wav e equatio n
(15)
wit h initia l condition t = 0 and u = e~ f «"2 (*/3A * ) . Figure 5 shows
the compute d results of the sevenpoint DRP scheme, the stan
dard fourth  and sixthorde r centra l differenc e schemes. The Fourier
transfor m of the initial data is a Gaussia n wit h the mai n part of the
spectrum lying in the range a Ax < 1.0. Thus the DRP scheme can
provid e adequat e resolutio n for this problem. The group velocit y of
the wav e component s in the range 0.8 < a Ax < 1.0 of the sixth 
order scheme is considerabl y less tha n #o. This par t of the com
puted wave s lags behind the mai n puls e as show n in Fig. 5. The
fourthorde r scheme becomes quite dispersive for a Ax > 0.6. The
compute d resul t exhibit s largeamplitud e trailin g waves. In general,
0.6
0.4
U 0.2
0.0
a)
b)
c)
0.4
U 0.2
0.0
0.2
390.0
X/&X
Fig. 5 Comparison between the computed and the exact solutions of
the simple onedimensional wave equation; ——, numerical solution;
.........9 exact solution: a) fourthorder central differenc e scheme, b)
sixthorder central difference scheme, and c) DRP scheme (sevenpoint
stencil).
loworde r scheme s are more likel y to be affecte d by numerica l
dispersion.
D. Artificial Selective Damping
To obtai n a highqualit y numerica l solution, it is necessar y to
eliminat e the shor t wavelengt h spuriou s numerica l waves. This can
be done by introducin g artificia l selective dampin g terms in the finit e
differenc e equations. The idea of usin g artificia l dampin g terms to
smoot h out the profil e of a shock is not new.13'14 Tarn et al.15 refin e
the idea by developin g a way to tailor the dampin g terms specificall y
for eliminating only the short waves. For their damping scheme, the
long wave s (a Ax < 1.0) are effectivel y untouched.
Conside r the linearize d wmomentu m equatio n discretize d on a
mesh of spacing AJC. Suppose a linear damping term consisting of
all of the values of u in the sevenpoin t stencil is added to the right
side of the equation. At the ith mesh point, the discretize d equatio n
may be writte n as
dt
(16)
j =3
where [va/( Ax)2] is the damping coefficient. The Fourier transfor m
ofEq. (16) is
dii
•D(aAx)u
where
D(aAx) =
eijuAx
(17)
(18)
Equation s (17) and (18) show tha t the amoun t of dampin g depends
on the wave number s so that by choosing the various dj properl y
one can damp onl y the shor t waves. A way to choose dj is propose d
in Refs. 12 and 15. In the Appendix, a set of value s of dj so obtaine d
is provided. The damping curve D(a Ax) vs a Ax is shown in Fig. 6.
There is practicall y no dampin g for long wave s (a Ax < 1.0). Also
shown in thi s figur e are the damping curves for the five  and three
poin t stencils. These smaller stencil s are needed at the boundar y
point s wher e a sevenpoin t stenci l woul d not fit.
To show the effectivenes s of the artificia l selective dampin g terms,
let us agai n conside r the numerica l solutio n of wave equatio n (15).
But thi s time, we choose a discontinuou s "boxcar" initia l condi 
tion, i.e.,
= 0,
u = Q.5[H(x + 50)  H(x  50)] (19 )
wher e H(x) is the uni t step function. Figure 7a shows the com
puted resul t at t = 200(Ajc/ao ) withou t artificia l dampin g terms.
0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
aAx
Fig.6 Damping function D(ctAx): ——, sevenpoint stencil (cr = 0.2?r);
— • —, sevenpoint stencil (a = 0.37r); .......... fivepoint stencil;
    , threepoint stencil.
TAM
0.4
U 0.2
0.4
U 0.2
I M M U..I M....M.I M M M M.I M.......I M..
\
U^^.1
/ —— \




,
300.0 200.0 100.0 0.0 100.0 200.0 300.0 400.0
x/Ax
1793
b)
Fig. 7 Waveform initiated by a disturbance with a boxcar profil e
showing parasite waves and the effec t of artificial selective damping:
a) no artificial damping and b) with artificial damping.
The spurious waves of the computed solution are generated by the
discontinuitie s of the initial condition. The gridtogri d oscillations
have the highest group velocity. They are found to the lef t of the
velocity pulse because their group velocity is negative (da/da < 0).
Figure 7b shows the computed solution with artificia l damping
terms (/?stenci l = 0.05). Obviousl y the spuriou s short wave s are
largely removed. The quality of the computed solution has greatly
improved.
IV. Radiation and Outflow Boundary Conditions
In the past, the subjec t of radiation boundar y conditions has been
studied by numerou s investigators. One group of investigator s used
asymptotic solutions to construct radiation boundar y conditions
These investigator s include Bayliss and Turkel,16'17 Hagstro m and
Hanharan,18 Harihara n et al.,19 and Tar n and Webb,6 to mentio n a
few. Anothe r group used the idea of characteristics. These investi 
gators include Thompson,2021 Giles,22 and Poinsot and Lele.23 Still
another group devised ways to construct absorbing boundary condi
tions to minimize the reflectio n of waves off the artificia l boundar y
of the computatio n domain. Investigator s of this group areEngquis t
and Majda,2425 Higdon,2627 Jiang and Wong,28 and Koslof f and
Kosloff. Recently, Givoli30 wrote a review article on this subject
with extensive references. However, the vast majorit y of the ref 
erenced works are devoted to the simple wav e equation. Since the
Euler equations, unlike the simple wave equation, support not only
acoustic but also vorticity and entropy waves, only a small subset
of the aforementione d reference s are relevan t to outflo w boundar v
conditions.
The formulatio n of radiation and outflo w boundar y condition s by
means of the asymptoti c solutions of the problem (strictl y speak
ing, they shoul d be the asymptoti c solutions of the finit e differenc e
equations 3 ) is quite straightforward. Here, the asymptoti c solutions
model the numerica l solution outside the computatio n domain. For
example, consider the problem of sound transmission through a one
dimensiona l variable area duct as shown in Fig. 8. Upstream of the
computatio n domain, it will be assumed that the duct has a constan t
area carryin g a subsoni c mean flo w u, pressure p, and densit y p The
fac t that the duct has constan t area implies the existence of asymp
totic solutions valid all of the way to x > oo. In this region the
governing equations are the linearized Euler equations:
pu f pu
uu+ 4
p
__pu + ypu _
= 0
(20)
Fig. 8 Schematic diagram of the computation domain for a one
dimensional flow in a variable area duct with constant area termina
tions. Inflow disturbances at the left boundary consist of sound and
entropy waves.
It is easy to show that the general solution of Eq. (20) is
J_
L F(^~')+ °
(pa) \a u J Q
1
1
(21)
wher e F, G, and # are arbitrar y function s and a = ( yp/p) l/2 is
the speed of sound. The solutions associated wit h the F and G func 
tions are the incoming acoustic and entropy waves. They are known
function s at the inflo w region. The solution associated with the H
functio n represents the reflecte d acoustic waves. It is not known a
priori. On eliminating H fromEq. (21), the followin g inflo w bound
ary conditions are derived:
(!**}
\auBt dxj
P
u
P.
1
a 2
1
(pa)
I
2a
T2
r,x
~~(jr I —
u(au) \u
(22)
In Eq. (22) F' and G' are the derivatives of F and G.
Now, let us return to the supersoni c jet noise problem of Fig. 1.
Radiation boundar y conditions, whic h allow sound wave s to prop
agate out of the computatio n domai n agains t the incoming entrain 
ment flow, as well as outflo w boundar y conditions, which permit
an arbitrar y combination of acoustic, entropy, and vorticit y waves
to leave the computatio n domai n smoothl y followin g the jet flow,
are needed. The difficult y here is that the mean flo w is nonuniform!
Tar n and Dong32 recently considered this problem. They proposed
the followin g radiation boundar y conditions for twodimensiona l
time domai n computations:
1
V(r,9)3t
P P
u — u
V — V
P pj
p ~" p
u — u
V — V
0 (23)
_ p — p
where (r, 6) are polar coordinates centered near the middle of the
computatio n domain; p, u, v, and p are the mean flo w quantitie s at
1794
TAM
the boundar y region; an d V(r, 0) i s relate d t o th e mea n flo w velocit y
V — (u, v) and the sound speed a by
[a2  (V • e9)2] *
(24)
Fo r th e outflow, the y propose d a se t of boundar y condition s tha t ac 
count s fo r mea n flo w nonuniformity. I f th e flo w i s unifor m Eq. (23 )
an d th e correspondin g outflo w boundar y condition s reduc e t o thos e
of Tar n an d Webb,6 whic h wer e derive d fro m th e asymptoti c so 
lution s of th e linearize d Eule r equation s by th e metho d of Fourie r
transform.
Recently, Hixo n et al.33 teste d computationall y th e effectivenes s
of th e radiatio n an d outflo w boundar y condition s of Thompson,20'21
Giles,22 an d Tar n an d Webb.6 Thei r findin g wa s tha t th e bound 
ar y condition s base d on asymptoti c solution s performe d well, bu t
th e characteristi c boundar y condition s produced significan t reflec 
tions. Other s als o reporte d simila r experience. I t i s worthwhil e t o
poin t ou t tha t fo r two  or threedimensiona l problems, ther e ar e no
genuin e characteristics. Wheneve r th e wave s inciden t obliquel y on
the boundar y o r ther e i s a significan t componen t o f mea n velocit y
paralle l t o th e boundary, th e validit y o f an y pseudocharacteristi c
formulatio n of boundar y condition s become s suspected. Grea t car e
shoul d be exercise d i n thei r usage.
V. Computation of Nonlinear Acoustic Waves
Nonlinearit y cause s the waveform of an acousti c puls e to steepe n
up an d ultimatel y t o for m a shock. I n th e stud y of Tar n an d Shen,12 i t
wa s foun d tha t th e nonlinea r wav e steepenin g process, whe n viewe d
in th e wav e numbe r space, corresponde d t o a n energ y cascad e
proces s whereb y lo w wav e numbe r component s ar e transferre d t o
the hig h wav e numbe r range. I f a highorde r finit e differenc e schem e
wit h a large bandwidt h of long waves (waves wit h a ~ a) in the
wav e numbe r spac e is use d fo r the computation, the compute d non 
linea r wavefor m remain s accurat e as long as th e cascadin g proces s
doe s no t transfe r wav e component s int o th e unresolve d (short ) wav e
numbe r range. Since, i n mos t aeroacousti c problems, th e soun d in 
tensit y i s no t sufficien t t o caus e th e formatio n of acousti c shocks,
the use of a highorde r finit e differenc e schem e suc h as the DRP
schem e woul d generall y be quit e adequate.
If shock s ar e formed, i t is know n tha t highorde r scheme s gener 
all y produc e spuriou s spatia l oscillation s aroun d the m an d in region s
wit h stee p gradients. Thes e spuriou s spatia l oscillation s ar e wave s i n
the shor t wav e (hig h wav e number ) rang e generate d by th e nonlinea r
wav e cascadin g process. Th e highorde r ENO 10 schem e wa s con 
ceive d an d designe d to hav e shockcapturin g capability. It shoul d be
Fig. 9 Computation of a nonlinear acoustic pulse using the DRP
scheme with variable artificia l damping; Rstendi = 0.05, t = 40Ajc/a0,
and Gaussian initial waveform with Mma x = 0o: —— > numerical solution;
.........9 exact solution.
the method of choic e fo r thi s typ e of problem. Th e EN O schemes,
however, automaticall y perfor m extensiv e testin g befor e a finit e dif 
ference approximation is applied. As a result, it is CPU intensive.
If the shock is not very strong, the more straightforwar d seven
point DRP scheme12 with artificia l selective damping terms added
to eliminat e th e spuriou s hig h wav e numbe r oscillation s aroun d
the shoc k ma y be use d instead. Fo r stron g shock s i t i s necessar y
to appl y the DRP scheme to the governin g equations writte n in
conservatio n form; otherwis e th e compute d shoc k spee d ma y no t
be accurate. Figur e 9 show s th e compute d wavefor m of an acous 
ti c puls e a t t — (40Ajt/0 () ) usin g th e sevenpoin t DR P schem e
wit h variabl e artificia l selectiv e damping.12 Initiall y th e puls e ha s
a Gaussia n wavefor m i n u wit h a halfwidt h equa l t o 12Ax. As
can be seen, th e compute d wavefor m compare s quit e wel l wit h th e
exac t solution. Th e shock, sprea d ove r fourfiv e mes h spacings, i s
no t a s shar p a s thos e obtaine d by usin g speciall y designe d shock 
capturin g schemes. Bu t thi s is to be expected.
VI. Wal l Boundary Conditions
It wa s pointe d ou t i n Sec. II.G tha t i f a highorde r finit e differ 
enc e schem e is use d to approximat e the governin g partia l differentia l
equations, the n th e numerica l solutio n i s boun d t o contai n spuriou s
components. Thes e spuriou s solution s ca n b e generate d b y initia l
conditions, nonlinearities, an d boundar y conditions. Fo r example, in
the reflectio n of acousti c wave s by a soli d wall, th e reflecte d wave s
woul d consis t of thre e distinc t components.34 Th e firs t componen t i s
the reflecte d wav e tha t closel y approximate s th e exac t solution. Th e
secon d componen t consist s of spuriou s shor t waves. Figur e 3 show s
tha t ther e ar e two (real ) value s of a fo r a give n a. Th e firs t com 
ponen t correspond s to a ~ a. The secon d componen t correspond s
to th e valu e a > ct. Th e third componen t i s mad e up of spatiall y
dampe d waves. The y correspon d to the comple x root s of a in the a
vs a relation. Fo r th e soun d reflectio n problem, thes e dampe d wav e
solution s ar e excite d by th e inciden t soun d wave s a t th e wall. Thei r
amplitude s deca y exponentiall y a s the y propagat e awa y fro m th e
wall. Effectively, they form a numerical boundar y layer adjacen t to
the wal l surface.
Figur e 1 0 show s th e mes h layou t fo r computin g th e soun d re 
flectio n problem. The wall is at y = 0. The interior points are
point s lyin g thre e or mor e row s awa y fro m th e wall. Thei r com 
putatio n stencil s li e entirel y insid e th e physica l domain. Th e firs t
thre e row s of point s adjacen t t o th e wal l ar e boundar y points. Thei r
sevenpoin t stencil s exten d outsid e the physica l domain. Th e point s
outsid e th e computatio n domai n ar e ghos t point s wit h n o obviou s
physica l meaning. However, Tar n an d Dong 34 observe d tha t ghos t
point s ca n b e usefu l fo r th e followin g reason. Recal l tha t th e so 
lutio n of th e Eule r or NavierStoke s equation s satisfie s th e partia l
differentia l equation s a t ever y interio r o r boundar y point. I n addi 
tion, a t a poin t o n th e wal l th e solutio n als o satisfie s th e appropriat e
boundar y conditions. No w th e discretize d governin g equation s ar e
no mor e tha n a se t of algebrai c equations. In th e discretize d system,
eac h flo w variabl e a t eithe r a n interio r or boundar y poin t i s governe d
by a n algebrai c equatio n (discretize d for m o f th e partia l differentia l
equations). Th e numbe r of unknown s is exactl y equa l to th e numbe r
of equations. Thu s ther e wil l be to o man y equation s an d no t enoug h
unknown s i f i t i s insiste d tha t th e boundar y condition s a t th e wal l
ar e satisfie d also. Thi s is, perhaps, on e of th e majo r difference s be 
twee n partia l differentia l equation s an d differenc e equations. Bu t
Wal l
uenor point s
boundar y point s
•* • y = 0
— ghos t point s
Fig. 10 Mesh layout adjacent to a plane wall showing the interior
points, boundary points, and ghost points.
TAM
1795
now the extra conditions imposed on the flo w variables by the wall
boundar y condition s can be satisfie d if ghost values are introduce d
(extra unknowns). The numbe r of ghos t value s is arbitrary, but the
minimu m numbe r mus t be equal to the numbe r of boundar y condi 
tions. Tar n and Dong suggested to use one ghos t value per bound 
ary point per physica l boundar y condition. To eliminat e the need
for extra ghos t values, they employed backwar d differenc e sten
cils to approximat e the spatia l derivative s at the boundar y points.
For the plane wal l problem, thei r analysi s indicate d that the pre
ceding wal l boundar y treatmen t woul d onl y give rise to ver y low
amplitud e spuriou s reflecte d waves. The thicknes s of the numerica l
boundar y layer was also ver y small regardles s of the angl e of inci
dence even when onl y six mesh point s per wavelengt h were used in
the computation.
In most aeroacoustic s problems, the wal l surfac e is curved. In
CFD, the standar d approac h is to map the physica l domai n int o a
rectangula r computationa l domai n wit h the curve d surfac e mappe d
into a plane boundar y or use unstructure d grids. For aeroacous 
tic problems, this is not necessaril y the best method. Mapping or
unstructure d grids effectivel y introduc e inhomogeneitie s int o the
governin g equations. Such inhomogeneitie s coul d cause unintende d
acousti c refractio n and scattering. An alternativ e way is to retai n a
Cartesia n mesh and to develop special treatment s for curve d walls.
Kurbatski i and Tarn35 developed one such treatmen t by extendin g
the one ghost value per boundar y point per physica l boundar y con
dition of Tar n and Dong.34 They tested thei r curve d wal l bound 
ary condition s by solvin g a series of linea r twodimensiona l acous
tic wave scattering problems. Morri s et al.36 propose d not to use
the wal l boundar y condition. Instead they simulate d the chang e in
impedanc e at the wal l by increasin g the densit y of the flui d inside
the solid body. At this time, it is too earl y to judge how well these
alternativ e method s woul d perfor m in problems wit h comple x wal l
boundaries. But for problems involvin g simpl e scatterer s suc h as cir
cular and elliptic cylinders, excellent compute d result s of the entired
scattered acousti c fiel d have been obtained.35 In any case, mappin g
or unstructure d grids ma y not be absolutel y necessar y for aero
acoustic s problems.
VII. Concluding Remarks
As a subdiscipline, CAA is still in its infancy. In thi s paper, some
of the relevan t computationa l issues and method s are discusse d (fo r
a set of benchmar k problems designe d to address some of these is
sues see Ref. 37). Obviously, the developmen t of new method s is
ver y much needed. However, it is also pertinen t to echo the belief
that application s of CAA to importan t or as yet unsolve d aeroacous 
tics problems are just as needed. It is necessar y to demonstrat e the
usefulness, reliability, and robustnes s of CAA. Unless and unti l thi s
is accomplished, CAA will remain merely a research subject but not
an engineerin g tool.
Appendix: Stencil and Damping Coefficients
The coefficient s of the sevenpoin t DRP scheme are
ao = 0 ai = a_! = 0.770882380518
a2 = a2 = 0.166705904415
a3 = a_3 = 0.208431427703
Backwar d stencil coefficient s ar e a"m, j = — n, — n +1,..., m — 1,
m (n — numbe r of point s to the lef t and m = numbe r of point s to
the right):
a™ = a™ = 2.192280339
ao6 = _a60 = 4.748611401
00 6 = _ fl (G » = _5.io8851915
of = a™3 = 4.461567104
af = a™4 = 2.833498741
af = fl «= 1.128328861
fl 06 _ _fl6o _ 0.203876371
al_\ = al1 = 0.209337622
alQ5 = al1 = 1.084875676
a}5 = a5_\ = 2.147776050
al25 = a5_\ = 1.388928322
a*5 = a5_\ = 0.768949766
al45 = a5_\ = 0.281814650
al55 = a^5 = 0.048230454
*2 = ~fl22 = 0.049041958
a2^ = af = 0.468840357
a™ = af = 0.474760914
a24 = a42! = 1.273274737
a24 = a422 = 0.518484526
a24 = a423 = 0.166138533
a24 = a424 = 0.026369431
The coefficient s of the fourleve l timemarchin g stenci l are
bo = 2.302558088838
bi = 2.491007599848
b2 = 1.574340933182
b3 = 0.385891422172
The coefficient s of the sevenpoin t dampin g stenci l are
(or = 0.27T) (a = Q.37T)
do = 0.287392842460 0.327698660845
di = <L i = 0.226146951809 0.235718815308
d2 = d.2 = 0.106303578770 0.086150669577
d3 = d_3 = 0.023853048191 0.014281184692
The coefficient s of the fivepoin t dampin g stencil are
d0 = 0.375
di = J_i = 0.25
d2 = d2 = 0.0625
The coefficient s of the threepoin t stenci l are
d0 = 0.5, di = di = 0.25
Acknowledgments
This wor k was supporte d by NAS A Lewi s Researc h Cente r Gran t
NAG 31267. Part of thi s wor k was writte n whil e the autho r was
in residenc e at the Institut e for Compute r Application s in Science
and Engineering. The autho r wishe s to than k Hao Shen and Davi d
Kopriv a for thei r assistance.
References
*Hubbard, H. H. (ed.), Aeroacoustics of Flight Vehicles: Theory and
Practice, Vol. 1: Noise Sources, Vol. 2: Noise Control, NAS A RP1258,
Aug. 1991.
2Tam, C. K. W., "Jet Noise Generate d by LargeScale Coheren t Mo
tion," Aeroacoustics of Flight Vehicles, NASA RP1258, Aug. 1991, Chap.
6, pp. 311390.
3Tam, C. K. W., "Supersonic Jet Noise," Annual Review o f Fluid Mechan
ics, Vol. 27,1995, pp. 1743.
4Seiner, J. M., "Advance s in HighSpeed Jet Aeroacoustics," AIAA Paper
842275, Oct. 1984.
5Roe, P. L., "Technical Prospect s for Computationa l Aeroacoustics,"
AIAA Paper 9202032, May 1992.
6Tam, C. K. W., and Webb, J. C., "DispersionRelationPreservin g Finite
Differenc e Schemes for Computationa l Acoustics," Journal of Computa
tional Physics, Vol. 107, Aug. 1993, pp. 262281.
7Thomas, J. P., and Roe, P. L., "Development of NonDissipativ e Numer 
ical Schemes for Computationa l Aeroacoustics," AIAA Paper 933382, Jul y
1993.
8Zingg, D. W., Lomax, H., and Jurgens, H., "An Optimize d Finite Differ 
ence Scheme for Wave Propagatio n Problems," AIAA Paper 930459, Jan.
1993.
9Lockard, D. P., Brentner, K. S., and Atkins, H. L., "High Accuracy
Algorithm s for Computationa l Aeroacoustics," AIAA Paper 940460, Jan.
1994.
10Harten, A., Engquist, A., Osher, S., and Chakravarthy, S., "Uniforml y
HighOrde r Accurac y Essentiall y NonOscillator y Schemes III," Journal of
Computational Physics, Vol. 71, Aug. 1987, pp. 231323.
11 Whitham, G. B., Linear and Nonlinear Waves, WileyInterscience, New
York, 1974.
1796
TAM
12Tam, C. K. W., and Shen, H., "Direc t Computatio n of Nonlinea r Acous 
tic Pulse s Usin g HighOrde r Finit e Differenc e Schemes," AIA A Pape r 93
4325, Oct. 1993.
13 Vo n Neumann, J., an d Richtmyer, R. D., "A Metho d fo r the Numerica l
Calculation of Hydrodynami c Shocks," Journal o f Applied Physics, Vol. 21,
Marc h 1950, pp. 232237.
14Jameson, A., Schmidt, W., an d Turkel, E., "Numerica l Solution s of
the Eule r Equation s by Finit e Volum e Method s Usin g RungeKutt a Time
Steppin g Schemes," AIA A Pape r 811259, Jun e 1981.
15Tam, C. K. W., Webb, J. C., and Dong, Z., "A Stud y of the Shor t Wav e
Component s in Computationa l Acoustics" Journal of Computational Acous
tics, Vol. 1, Marc h 1993, pp. 130.
16Bayliss, A., an d Turkel, E., "Radiatio n Boundar y Condition s for Wave 
Like Equations," Communications on Pure an d Applied Mathematics, Vol.
33, Nov. 1980, pp. 707725.
17Bayliss, A., an d Turkel, E., "Far Fiel d Boundar y Condition s for Com 
pressibl e Flows," Journal o f Computational Physics, Vol. 48, Nov. 1982, pp.
182199.
18Hagstrom, T., an d Hariharan, S. L, "Accurat e Boundar y Condition s fo r
Exterio r Problem s in Ga s Dynamics," Mathematics o f Computation, Vol. 51,
Oct. 1988, pp. 581597.
19Hariharan, S. I., Ping, Y., and Scott, J. C., "Time Domai n Numerica l
Calculation s of Unstead y Vortica l Flows Abou t a Flat Plat e Airfoil," Journal
o f Computational Physics, Vol. 101, Aug. 1992, pp. 419430.
20Thompson, K. W, "Time Dependen t Boundar y Condition s for Hyper 
boli c Systems," Journal o f Computational Physics, Vol. 68, Jan. 1987, pp.
124.
21 Thompson, K. W., "Time Dependen t Boundar y Condition s for Hyper 
boli c Systems, II," Journal of Computational Physics, Vol. 89, Aug. 1990,
pp. 439^61.
22Giles, M. B., "Nonreflectin g Boundar y Condition s fo r Eule r Equatio n
Calculations," AIAA Journal, Vol. 28, No. 12, 1990, pp. 20502058.
23Poinsot, T. J., an d Lele, S. K., "Boundar y Condition s for Direc t Simu 
lation s of Compressibl e Viscou s Flows," Journal o f Computational Physics,
Vol. 101, Jul y 1992, pp. 104129.
24Engquist, B., an d Majda, A., "Radiatio n Boundar y Condition s fo r
Acousti c an d Elasti c Wav e Calculations," Communications on Pure an d
Applied Mathematics, Vol. 32, Ma y 1979, pp. 313357.
25Engquist, B., an d Majda, A., "Absorbin g Boundar y Condition s fo r th e
Numerical Simulation of Waves," Mathematics of Computation, Vol. 31,
Jul y 1977, pp. 629651.
26Higdon, R. L., "Absorbing Boundar y Conditions for Differenc e Ap
proximations to the MultiDimensiona l Wave Equation," Mathematics of
Computation, Vol. 47, Oct. 1986, pp. 629651.
27Higdon, R. L., "Numerica l Absorbin g Boundar y Condition s fo r
the Wave Equation," Mathematics of Computation, Vol. 49, Jul y 1987,
pp. 6590.
28 Jiang, H., an d Wong, Y. S., "Absorbin g Boundar y Condition s for Secon d
Orde r Hyperboli c Equations," Journal of Computational Physics, Vol. 88,
Ma y 1990, pp. 205231.
29 Kosloff, R., an d Kosloff, D., "Absorbin g Boundarie s fo r Wav e Propa 
gatio n Problems," Journal of Computational Physics, Vol. 63, Apri l 1986,
pp. 363376.
3() Givoli, D., "NonReflectin g Boundar y Conditions," Journal o f Compu
tational Physics, Vol. 94, Ma y 1991, pp. 129.
31 Tarn, C. K. W., an d Webb, J. C., "Radiatio n Boundar y Conditio n an d
Anisotrop y Correctio n fo r Finit e Differenc e Solution s of th e Helmholt z
Equation," Journal o f Computational Physics, Vol. 113, Jul y 1994, pp.
122133.
32Tam, C. K. W, an d Dong, Z., "Radiatio n an d Outflo w Boundar y Con 
dition s fo r Direc t Computatio n o f Acousti c an d Flo w Disturbance s i n a
Nonunifor m Mea n Flow," AIA A Pape r 95007, Jun e 1995.
33Hixon, D. R., Shih, S., an d Mankbadi, R. R., "Evaluatio n of Boundar y
Condition s fo r Computationa l Aeroacoustics," AIA A Pape r 950160, Jan.
1995.
34Tam, C. K. W, and Dong, Z., "Wal l Boundar y Condition s for High 
Orde r Finit e Differenc e Scheme s i n Computationa l Aeroacoustics," Theo
retical an d Computational Fluid Dynamics, Vol. 8, No. 6,1994, pp. 303322.
35 Kurbatskii, K. A., an d Tarn, C. K. W, "Curve d Wal l Boundar y Condi 
tion s fo r HighOrde r Finit e Differenc e Scheme s in Acousti c Wav e Scatterin g
Problems," Bulletin o f th e American Physical Society, Vol. 39, Nov. 1994,
p.1907.
36Morris, P. J., Chung, C., Chyczewski, T. S., an d Long, L. N., "Computa 
tiona l Aeroacoustic s Algorithms: Nonunifor m Grids," AIA A Pape r 942295,
Jun e 1994.
37 Hardin, J. C. (ed.), Computational Aeroacoustics Benchmark Problems,
Proceeding of the ICASE/LaRC Workshop on Benchmark Problems in Com
putational Aeroacoustics (Hampton, VA), NAS A CP3300, Ma y 1995.
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