Computational Aeroacoustics: Issues and Methods

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AIAA JOURNA L
Vol. 33, No. 10, October 1995
Computational Aeroacoustics: Issues and Methods
Christophe r K. W. Tarn *
Florida State University, Tallahassee, Florida 32306-3027
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 95-067 7 at th e AIA A 33r d
Aerospac e Science s Meeting, Reno, NV, Jan. 9-12,1995; 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 shock-associate 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 large-scal e turbulenc e structures/instabilit y
wave s and the fine-scal e turbulence. Bot h the large turbulenc e struc -
ture s and the fine-scal e turbulenc e are nois e sources. However, it is
known 2-3 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 fine-scal e turbulence, bein g less impor -
tant, is ignored. The resolutio n of fine-scal e turbulenc e is, therefore,
not a primar y issue.
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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 far-fiel 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 high-speed 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 noise-producin 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
sound-absorbing 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 far-fiel d noise spectrum of an imperfectly expanded su-
personic jet, measured at 30-deg 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 shock-associate 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 shock-associate 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 root-mean-squar 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 small-amplitud 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 large-magnitud e
disparit y issue.
C. Distinc t and Well-Separate 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 half-widt 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 low-orde r finit e differenc e schemes do not satisf y the preced-
ing condition.
Finite differenc e schemes, invariably, have built-i 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 Courant-Friedrichs-Lewy 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 time-marchin 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 time-independen 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 small-amplitud 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 three-dimensiona 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 shock-capturin 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 shock-capturin 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
shock-containin 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 first-orde r spatia l derivative s
of the Euler equation s ar e approximate d by first-orde 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
high-orde 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 high-orde 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 high-orde 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 high-orde 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 high-orde r essentiall y non-
oscillator y (ENO)10 and the dispersion-relation-preservin 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.
TAM
1791
A. Wave Number of a Finite Difference Scheme
Suppose a seven-point 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 a-j 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 sixth-order
standard central difference s cheme,..........
sixth-orde 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:
(n-j)
(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
— = ±00-7-
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 seven-poin 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 (grid-to-gri 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 seven-point DRP scheme, the stan-
dard fourth - and sixth-orde 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
fourth-orde r scheme becomes quite dispersive for a Ax > 0.6. The
compute d resul t exhibit s large-amplitud 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 one-dimensional wave equation; ——, numerical solution;
.........9 exact solution: a) fourth-order central differenc e scheme, b)
sixth-order central difference scheme, and c) DRP scheme (seven-point
stencil).
low-orde r scheme s are more likel y to be affecte d by numerica l
dispersion.
D. Artificial Selective Damping
To obtai n a high-qualit 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 w-momentu 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 seven-poin 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) =
e-ijuAx
(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 seven-poin 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 "box-car" 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): ——, seven-point stencil (cr = 0.2?r);
— • —, seven-point stencil (a = 0.37r); .......... five-point stencil;
- - - - -, three-point 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 box-car 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 grid-to-gri 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,20-21 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,24-25 Higdon,26-27 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:
(-!-*-*-}
\a-uBt dxj
P
u
-P.
1
a 2
1
(pa)
I
2a
T2
r,x
~~(jr I —
u(a-u) \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 two-dimensiona 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 three-dimensiona 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 high-orde 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 high-orde 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 high-orde 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 high-orde r ENO 10 schem e wa s con -
ceive d an d designe d to hav e shock-capturin 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 seven-poin 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 half-widt 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 four-fiv 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 high-orde 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
seven-poin 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 Navier-Stoke 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 two-dimensiona 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 seven-poin t DRP scheme are
ao = 0 ai = -a_! = 0.770882380518
a2 = -a-2 = -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 four-leve l time-marchin g stenci l are
bo = 2.302558088838
bi = -2.491007599848
b2 = 1.574340933182
b3 = -0.385891422172
The coefficient s of the seven-poin 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 five-poin t dampin g stencil are
d0 = 0.375
di = J_i = -0.25
d2 = d-2 = 0.0625
The coefficient s of the three-poin t stenci l are
d0 = 0.5, di = d-i = -0.25
Acknowledgments
This wor k was supporte d by NAS A Lewi s Researc h Cente r Gran t
NAG 3-1267. 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
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