NASA Technical Reports Server (NTRS) 20040110264: An Analytical Comparison of the Acoustic Analogy and Kirchhoff Formulation for Moving Surfaces

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by  NASA Technical Reports Server (NTRS)

AN ANALYTICAL COMPARISON OF THE ACOUSTIC ANALOGY AND KIRCHHOFF
FORMULATION FOR MOVING SURFACES
Kenneth S. Brentner and F. Farassat
Research Engineer Senior Research Scientist
NASA Langley Research Center
Hampton, Virgina
Abstract The FW{H equation is an exact rearrangement of
the continuity equation and the Navier{Stokes equa-
The Lighthill acoustic analogy, as embodied in the
tions into the form of an inhomogenous wave equa-
FfowcsWilliams{Hawkings(FW{H)equation,iscom-
tion with two surface source terms, known as thick-
pared with the Kirchho(cid:11) formulation for moving sur-
ness and loading sources, and a volume source term
faces. A comparison of the two governing equations
(thequadrupolesourcefromtheoriginalLighthillthe-
reveals that the main Kirchho(cid:11) advantage (namely
ory). Although the quadrupole source contribution is
nonlinear (cid:13)ow e(cid:11)ects are included in the surface in-
insigni(cid:12)cant in many subsonic applications, substan-
tegration)isalsoavailabletotheFW{Hmethodifthe
tiallymorecomputationalresourcesareneededforvol-
integration surface used in the FW{H equation is not
umeintegrationwhenthequadrupolesourcerequired.
assumed impenetrable. The FW{H equation is ana-
TheKirchho(cid:11)formulationformovingbodiesisalsoan
lytically superior for aeroacousticsbecause it is based
inhomogenous wave equation with source terms dis-
upon the conservation laws of (cid:13)uid mechanics rather
tributedonasurfacewhichenclosesallofthephysical
than the wave equation. This means that the FW{H
sources. The Kirchho(cid:11) formulation is attractive be-
equation is valid even if the integration surface is in
cause no volume integration is necessary.
the nonlinear region. This is demonstrated numeri-
cally in the paper. The Kirchho(cid:11) approach can lead
Althoughitisusefultohavemorethanoneformula-
to substantial errors if the integration surface is not
tionavailabletopredictnoise,thereisnoclearconsen-
positioned in the linear region. These errors may be
sus of which to choose for a particular application. A
hard to identify. Finally, new metrics based on the 5
recentnumericalcomparisonbyBrentneretal. ofthe
Sobolev norm are introduced which may be used to 6{8
helicopterrotornoisepredictioncodeWOPWOP+,
compare input data for both quadrupole noise calcu-
which uses a FW{H based formulation including an
lations and Kirchho(cid:11) noise predictions.
approximate quadrupole calculation, with a rotating
9,10
Kirchho(cid:11) code RKIR has shown that both meth-
Introduction
ods can predict the rotor noise equally well. In that
A great deal of progress has been made in recent work, however, neither method was demonstrated to
years toward the prediction of rotating-blade noise be clearly superior.
throughmethodsutilizing(cid:12)rstprinciples. Severalrea-
sons account for this progress. First, a detailed and The main purpose of this paper is to analytically
fundamental understanding of how rotor blades gen- compare these two acoustic prediction methodologies
erate noise has been gained through several acoustic and reduce the confusion that currently exists about
wind-tunnel and(cid:13)ight tests. Secondly, arigorousthe- the relationship between the two methods. This in-
oreticalbasisforpredictingnoisegeneratedbyrotating cludes a comparison of how the governing equations
blades has been developed. In fact, several prediction are derived, highlighting the di(cid:11)erences in the deriva-
methodologies with a solid physical and mathemati- tions. Both analytical comparison and numerical
cal basis are currently available: formulations based comparisons are necessary to determine whether one
upontheLighthillacousticanalogy1 (inparticularthe method has an advantage in terms of e(cid:14)ciency, ac-
Ffowcs Williams and Hawkings (FW{H) equation2) curacy, and robustness over the other. An alternate
and Kirchho(cid:11) formulations for both subsonic and su- implementation of the FW{H equation is presented
personic moving surfaces.3,4 whichcombinestheadvantagesofboththetraditional
formulation of the FW{H equation and the Kirchho(cid:11)
Presented attheAmerican HelicopterSociety53rd Annual Fo- formulation. Finally, a useful metric for comparing
rum, Virginia Beach, Virginia, April 29 -May 1, 1997. formulations will be outlined.
1
Advantages and Disadvantages in the linear (cid:13)ow region, such that the input acoustic
0 0
It is important to brie(cid:13)y consider the advantages pressure p (cid:17) p (cid:0)po and its derivatives @p=@t and
0
@p=@n are compatible with linear wave propagation.
and disadvantages of both the FW{H and Kirchho(cid:11)
Thelocationofthelinearregionisnotwellde(cid:12)nedand
formulations at this point in order to understand the
is problem dependent. It would be desirable to place
motivation for a more in depth analysis.
theKirchho(cid:11)surfacewellawayfromthesourceregion,
FW{H Description buttypicallyCFDsolutionsarenotaswellresolvedor
The FW{H approach has several advantages over asaccurateawayfromthebody. Hence,theplacement
theKirchho(cid:11)method. First,thethreesourcetermsin of the Kirchho(cid:11) surface is usually a compromise.
theFW{Hequationeachhavephysicalmeaningwhich
is helpful in understanding the noise generation. The Analytical Comparison
thicknessnoiseisdeterminedcompletelybythegeom-
Now that the general characteristics of both the
etry and kinematics of the body. The loading noise is
FW{H and Kirchho(cid:11) formulations have been de-
generated by the force acting on the (cid:13)uid due to the
scribed, a more detailed comparison will be helpful.
presence of the body. The classi(cid:12)cation of thickness
First, we shall consider the development of the gov-
and loadingnoise is related to the thickness and load-
erning equations of both approaches to gain insight
ing problems of linearized aerodynamics. Thus, this
into the validity of each type of formulation. Then
terminology is consistent with that of aerodynamics.
an assessment of an integral formulation for subsonic
Thequadrupolesourcetermaccountsfornonlinearef-
source motion will be considered.
fects(i.e.,nonlinearwavepropagationandsteepening,
variations in the local sound speed, noise generated Governing Equations
by shocks, vorticity, and turbulence in the (cid:13)ow (cid:12)eld, FW{H Equation
etc.)11{13 TheFW{Hequation2isthemostgeneralformofthe
All three sourceterms are interdependent, yet their Lighthill acoustic analogy and is appropriate for pre-
physical basis provides information to design quieter dicting the noise generated by the complex motion of
rotors. The separation of source terms also is an ad- helicopterrotors. TheFW{Hequationmaybederived
vantage numerically because not all terms must be by embedding the exterior (cid:13)ow problem into a prob-
computed at all times if it is known that a particu- leminunboundedspacebyusinggeneralizedfunctions
lar sourcedoes not contribute to the sound (cid:12)eld (e.g., to describe the (cid:13)ow (cid:12)eld. To do this, consider a mov-
for low-speed (cid:13)ow the quadrupole may be neglected, ing surface f(x;t)=0 with a stationary (cid:13)uid outside.
in the rotor plane thickness noise is dominant, etc.). The surfacef =0is de(cid:12)ned suchthat rf =n^, where
A(cid:12)naladvantageofFW{Hbasedformulationsisthat n^ is a unit normal vector pointing into the (cid:13)uid. In-
these formulations are relatively mature and have ro- sidef =0thegeneralized(cid:13)owvariablesarede(cid:12)nedto
bust numerical algorithms. The main disadvantage of have their freestream values, i.e.,
the FW{Hmethod isthatto predict thenoiseofbod- (cid:26)
(cid:26) f >0
ies moving at transonic speeds the quadrupole source (cid:26)~ = (1)
(cid:26)o f <0
must be included. This is a disadvantage because the (cid:26)
quadrupole|whichisavolumesource|ultimatelyre- (cid:26)ui f >0
(cid:26)fui = (2)
quiresavolumeintegrationoftheentiresourceregion. 0 f <0
Volume integration is computationally expensive and and (cid:26)
canbe di(cid:14)cult toimplement. Although the computa- P~ij = Pij f >0 (3)
tional e(cid:11)ort can be reduced by approximation of the 0 f <0
7,8
quadrupole, it cannot be avoided completely.
where the tilde indicates that the variable is a gener-
Kirchho(cid:11) Description alized function de(cid:12)ned throughout all space. On the
TheKirchho(cid:11)approachdoesnotsu(cid:11)erfromthispit- righthandside(cid:26),(cid:26)ui,andPij arethedensity,momen-
fall because it only has surface source terms. Hence, tum,andcompressivestresstensor,respectively. Note
the Kirchho(cid:11) method has been used for the past sev- that we have absorbed the constant (cid:0)po(cid:14)ij into the
eral years for the prediction of transonic rotor noise. de(cid:12)nition ofPij forconvenience,hence,foraninviscid
0
Unlike the FW{H source terms, however, the Kirch- (cid:13)uid, Pij =p(cid:14)ij. Freestream quantities are indicated
ho(cid:11) source terms are not easily related to thickness, by the subscript o and (cid:14)ij is the Kroneckerdelta.
loading,nonlineare(cid:11)ects,orindeedanyphysicalmech- Using de(cid:12)nitions (1){(3), a generalized continuity
anisms. They provide little guidance for design. An- equation can be written
otherdisadvantageoftheKirchho(cid:11)methodisthatthe @(cid:22)(cid:26)~ @(cid:22)(cid:26)fui 0@f @f
sourcesurface(Kirchho(cid:11)surface)mustbechosentobe + =((cid:26) +(cid:26)ui )(cid:14)(f) (4)
@t @xi @t @xi
2
where the bar over the derivative operators indicate FW{H equation. The di(cid:11)erence is that the domain
that generalized di(cid:11)erentiation (i.e., di(cid:11)erentiation of is now considered in terms of wave propagation. The
0
generalizedfunctions)isimpliedand(cid:26) (cid:17)(cid:26)(cid:0)(cid:26)o. Also surface f = 0 is de(cid:12)ned such that all of the acoustic
note that @f=@t = (cid:0)vn, @f=@xi = n^i and (cid:14)(f) is sources are contained inside the surface. Then, the
0
the Dirac delta function. This generalized continuity acoustic pressure p(x;t) is extended such that
equationis validfortheentirespace|bothinsideand (cid:26)
0
outsideofthebody. Thegeneralizedmomentumequa- pe0 = p f >0 (7)
tion can be written 0 f <0
@(cid:22)(cid:26)fui @(cid:22)(cid:26)guiuj @(cid:22)P~ij and the generalized wave equation|which is the
+ + = governing equation for the Kirchho(cid:11) formulation|
@t @xj @xj
becomes
@f @f
((cid:26)ui@t +((cid:26)uiuj +Pij)@xj)(cid:14)(f) : (5) 2 0 (cid:0)@p0Mn @p0(cid:1)
p(x;t) = (cid:0) + (cid:14)(f)
@t c @n
Nowbytakingthe timederivativeofequation(4)and @ (cid:0) 0Mn (cid:1) @ (cid:0) 0 (cid:1)
subtracting the divergence of equation (5), followed (cid:0) @t p c (cid:14)(f) (cid:0) @xi pn^i(cid:14)(f)
with some rearranging, the FW{H equation may be
(cid:17) Qkir (8)
writtenasthefollowinginhomogeneouswaveequation:
0
whereMn =vn=c. Inthisequationisp mustbecom-
2 0 @(cid:22)2 patible with the wave equation, hence, equation (8) is
p(x;t) = [TijH(f)]
@xi@xj valid only in the region of the (cid:13)uid in which the wave
@ equation is the appropriate governingequation.
(cid:0) [(Pijn^j +(cid:26)ui(un(cid:0)vn))(cid:14)(f)]
@xi Source Term Comparison
@ It is well known that the wave equation can be de-
+ [((cid:26)ovn+(cid:26)(un(cid:0)vn))(cid:14)(f)] (6)
@t rived directly from the conservation laws of (cid:13)uid me-
chanics, but it is our objective in this paper to show
where Tij is the Lighthill stess tensor, un is the (cid:13)uid
how equation (8) is related to the FW{H equation,
velocity in the direction normal to the surface f = 0
equation(6). Tothat end, weadd and subtractterms
andvnisthesurfacevelocityinthedirectionnormalto
to the inviscid form of equation (6) to manipulate the
the surface. On the left hand side we use the custom-
0 2 0 sourcetermsintotheformofequation(8). Thisyields
arynotationp (cid:17)c (cid:26) becausetheobserverlocationis
outside of the source region. 2 0 @(cid:22)2
UsuallyinthederivationoftheFW{Hequationthe p(x;t)=Qkir+ [TijH(f)]
@xi@xj
psuhryfsaiccealfbo=dy0 sisurafasscuemaenddtiombpeenceotirnacbildeen(utnwi=thvtnh)e. + (cid:0)@p0Mn + @p0(cid:1)(cid:14)(f)+ @ (cid:2)(p0(cid:0)c2(cid:26)0)Mn(cid:14)(f)(cid:3)
@t c @n @t c
That assumption is not necessary and has not been
@ (cid:2) (cid:3) @ (cid:2) (cid:3)
madeinequation(6)sothatitmaybecomparedmore (cid:0) (cid:26)ui(un(cid:0)vn)(cid:14)(f) + (cid:26)un(cid:14)(f) : (9)
directly with the governing equation of the Kirchho(cid:11) @xi @t
formula for moving surfaces. Ffowcs Williams and
If we note that
Hawkingsused slightlydi(cid:11)erent mathematicalmanip-
ulations, but it is clear from their paper2 that they @(cid:22)2H(f) @ (cid:0) (cid:1) @ (cid:0) (cid:1)
= n^i(cid:14)(f) =(cid:0) vn(cid:14)(f) (10)
understood it is not essential to choose the integra- @t@xi @t @xi
tion surface coincidental with the physical body. Re-
14 15 and utilize the continuity and momentum equations
cently di Francesantonio and Pilon and Lyrintzis
we can rewrite equation (9) as
have also treated the FW{H on a permeable surface,
but have used di(cid:11)erent names to identify the form 2 0 @(cid:22)2
of the FW{H equation given in equation (6). (Pilon p(x;t)=Qkir+ [TijH(f)]
@xi@xj
and Lyrintzis results appear to be incorrect because
theyhavesubstituedp0 forc2(cid:26)0 insomeoftheirsource + @ [p0(cid:0)c2(cid:26)0]Mn(cid:14)(f)+ @ (cid:2)(p0(cid:0)c2(cid:26)0)Mn(cid:14)(f)(cid:3)
@t c @t c
terms.)
@ @
Kirchho(cid:11) Equation (cid:0) [(cid:26)uiuj]n^i(cid:14)(f)(cid:0) [(cid:26)uiun(cid:14)(f)] : (11)
@xj @xi
The development of the Kirchho(cid:11) formulation, due
3
to Farassat and Myers, utilizes the same mathemat- This form of the FW{H equation is helpful because
ical style and rigor as used in the derivation of the the source terms that are not found in the Kirchho(cid:11)
3
governing equation are easily identi(cid:12)ed. This is an has two pitfalls: it is not easily recognized as the
important result of this paper. All of the additional FW{H equation, and there are no clear connections
source terms are second order and may be neglected betweentheformofthesourcetermsandtheproblem
inthelinear(cid:13)owregion. ThiswaspreciselyLighthill’s physics.
originalpremise|thewaveequationistheappropriate
governing equation outside of a limited source region. An Integral Formulation
0 2 0
In fact, when p =c (cid:26) equation (11) becomes NowthattherelationshipbetweentheFW{Hequa-
tionandtheKirchho(cid:11)formulationhasbeendeveloped
2p0(x;t)=Qkir+ @(cid:22)2(cid:26)uiujH(f) : (12) on the governing equation level, we would like to de-
@xi@xj velopanapplicableintegralform whichis appropriate
forsubsonicsourcemotion. Thisisneededforultimate
Noticethatthe Heavisidefunction hasbeen takenout
implementation and numerical comparison of the dif-
oftheequation(11)quadrupolesourceterminthema-
ferent formulations.
nipulations leading to equation (12). The only source
term remaining which is not in equation (8) is clearly A slightly modi(cid:12)ed integral formulation for the
second order in the perturbation quantity ui. This FW{H equation is needed because the current prac-
term would be neglected in the derivation of the wave tice is to assume that the FW{H integration surface
equation from the (cid:13)uid conservation laws. Hence, we corresponds to the body and is impenetrable. Equa-
have shown that the FW{H and Kirchho(cid:11) formula- tion(6)istheappropriateformoftheFW{Hequation
tions are indeed equivalent when the integration sur- to start the development of an integral representation
face for both is placed in the linear region of the (cid:13)ow whichhasthesameformasthetraditionalapplication
14
(i.e.,wheretheinputdataiscompatiblewiththewave oftheFW{Hequation. FollowingdiFrancesantonio,
equation). we de(cid:12)ne new variables Ui and Li as
TheFW{HequationandtheKirchho(cid:11)arequitedif-
(cid:26) (cid:26)ui
ferent, however,when the integration surfaceis in the Ui = (1(cid:0) )vi+ (14)
source region. The implications of this di(cid:11)erence is (cid:26)o (cid:26)o
demonstrated later with numerical examples. If the and
FW{H equation integration surface is on the body Li = Pijn^j +(cid:26)ui(un(cid:0)vn) : (15)
or in the source region, the quadrupole|a volume
source term|must be included to accurately predict We have chosen a slightly di(cid:11)erent, but equivalent,
thenoise. Therefore,wecaninferthataswemovethe de(cid:12)nitionsfromthatofreference14because(cid:26)and(cid:26)ui
integration surface of the FW{H equation away from areconservationvariablesoftenutilizedinCFDcodes.
the body, the contribution of the volume quadrupole With these de(cid:12)nitions, the FW{H equation may be
contained within the surface must now be accounted written in its standard di(cid:11)erential form:
for by the surface source terms. We shall numerically
demonstrate this later. 2 0 @2
Forcompleteness,equation(11)canbesimpli(cid:12)edby p(x;t)= [TijH(f)]
@xi@xj
canceling terms and rearranging. The result is
@ @
2 0 (cid:0)@c2(cid:26)0Mn @(cid:26)ui (cid:1) (cid:0) @xi[Li(cid:14)(f)]+ @t[((cid:26)oUn)(cid:14)(f)] : (16)
p(x;t)=(cid:0) + n^i (cid:14)(f)
@t c @t
@ (cid:0) 2 0Mn (cid:1) @ (cid:2) 0 (cid:3) Thisequationis particularlyusefulbecauseFarassat’s
(cid:0)@t c (cid:26) c (cid:14)(f) (cid:0) @xi (pn^i+(cid:26)uiun)(cid:14)(f) formulation1A6,16 canbeutilizeddirectlytowritean
2 integral representation of the solution as
@
+ [TijH(f)] : (13)
@xi@xj 0 0 0 0
p(x;t)=pT(x;t)+pL(x;t)+pQ(x;t) (17)
Notice that the surface source terms in equation (13)
are closely related to equation (8). In fact by sub- where
2 0 0
stituting c (cid:26) for p in the time derivative terms in
0 Z
equation(8)and(cid:26)uiuj+p(cid:14)ij inthespatialderivative 0 (cid:2)(cid:26)o(U_n+Un_)(cid:3)
terms we can get the surface source terms in equa- 4(cid:25)pT(x;t)= r(1(cid:0)Mr)2 retdS
tion (13). (The momentum equation was used to ex- f=0
change @((cid:26)uiuj +p0(cid:14)ij)=@xj with (cid:0)@(cid:26)ui=@t in equa- Z (cid:2)(cid:26)oUn(rM_r+c(Mr(cid:0)M2))(cid:3)
tion (13).) While the correspondence between equa- + dS ;
r2(1(cid:0)Mr)3 ret
tion(13)andequation(8)isinteresting,equation(13) f=0
4
Z
0 1 (cid:2) L_r (cid:3) velopedtotestthenumericalimplementationofequa-
4(cid:25)pL(x;t)= c r(1(cid:0)Mr)2 retdS tion (17) without the quadrupole source term. The
f=0 modi(cid:12)ed code is called FW{H/RKIR in this paper.
Z
(cid:2) Lr(cid:0)LM (cid:3) RKIR was chosen as the platform to test the new
+ dS
r2(1(cid:0)Mr)2 ret FW{H implementation primarily because it already
f=0
performs integration on a surface some distance from
Z
1 (cid:2)Lr(rM_r+c(Mr(cid:0)M2))(cid:3) a rotor blade and has been coupled to the full poten-
+ c r2(1(cid:0)Mr)3 retdS ; tial (cid:13)ow solver FPRBVI.18,19 A third code, WOP-
f=0 WOP+,8 which utilizes the traditional FW{H imple-
0 mentation (surface integration on the blade surface
and pQ(x;t) can be determined by any method cur-
and an approximate quadrupole implementation) will
rentlyavailable(e.g.,seereference8). Inequation(17)
also be used in the comparison.
the dotoveravariableimpliessourcetime di(cid:11)erentia-
tionofthatvariable,LM =LiMi,andasubscriptr or 100
n indicates a dot product of the vector with the unit
vector in the radiation direction ^r or the unit vector
in the surface normal direction n^, respectively. 0
Current rotor noise prediction codes can easily be
modi(cid:12)ed to accommodate this new implementation of
the FW{H equation. The major di(cid:11)erence is that the -100 Kirchhoff
integration surface is no longer restricted to the rotor FW-H
blade surface and in addition to p0, the values of (cid:26), p¢, Pa
data
(cid:26)ui are needed as input. When the surface does cor- -200
respond to the blade surface, the separation of source
terms into thickness, loading, and quadrupole noise
still has physical meaning; otherwise, the separation -300
0 0 0
of the source terms into pT, pL, and pQ is only math-
ematical. Hence, the ability to give physical interpre-
tation to the source terms continues to be a distinct -400
and unique advantage of the FW{H equation. 0.0 0.5 1.0 1.5 2.0
time, msec
Numerical Comparison of Formulations
Although we have shown analytically that the
FW{H formulation has advantages over the Kirchho(cid:11)
20
Figure 1. Comparison of predicted and measured
formulation, what really matters is how they com-
acoustic pressure at an in-plane observer location,
pareinpractice. Somecomparisonshavealreadybeen
3:4RfromtherotorhubofanuntwistedUH-1Hmodel
made(e.g.,seereferences5,14and17). Inreference14,
di Francesantonioconcluded that the main advantage rotor in hover (MH =0:88).
oftheFW{H equationappliedonaKirchho(cid:11)-typein-
tegration surface is that interaction with CFD codes The (cid:12)rst comparison is for an untwisted UH-1H
is easier because the normal derivative of pressure is model-scale rotor operating in hover with a hover-tip
20
no longer required. If this is the only advantage, and Mach number MH = 0:88. Figure 1 shows a com-
indeed we recognize that the normal derivative calcu- parison of acoustic pressure time history for both the
lationcan becumbersome,a simplesolutionwouldbe Kirchho(cid:11)andFW{Hmethodsonaintegrationsurface
to make the substitution which was located approximately 1.37 chords away
from the rotor in the direction normal to the blade
@p @(cid:26)ui
=(cid:0)n^i (18) surface and extending 1.25 chords beyond the blade
@n @t
tip. The full potential computation wasperformed on
in equation (8). This result is just the linear momen- a 80(cid:2)36(cid:2)24 grid, which is somewhat coarse. The
tum equation, which is applicable in the linear (cid:13)ow two computations are almost indistinguishable in this
region. Nevertheless, we believe there are other ad- case|an indication that the integration surface is in-
vantages which we will now demonstrate numerically. deed in the linear (cid:13)ow region. The underprediction
Forthiswork,anewcomputercodebasedonamod- of the negative peak is a result of using a coarse grid.
5
i(cid:12)cation of the RKIR code (Rotating KIRchho(cid:11) for- Brentneret al. found thatthe agreementisimproved
10
mulation) developedbyLyrintziset al. hasbeen de- with a(cid:12)ner grid. Small oscillations in the signal, near
5
0
k=2
-1000
k=7
k=12
-2000
k=18
p¢, Pa
k=21
-3000
data
-4000
-5000
0.0 0.5 1.0 1.5 2.0
time, msec
Figure 2. Cross section showing the location of the
Figure 3. Comparison of predicted acoustic pressure
integration surfaces with respect to the rotor blade.
using the Kirchho(cid:11) formulation with varying integra-
Theverticaldistancefromthebladechord, inunits of
tion surface locations. These predictions are for an
chord length, are labeled z=c. The value of the grid
observerlocated3:4RfromaUH-1Hmodelrotorhov-
index normal to the blade is labeled k.
ering at MH = 0:88. The experimental data is from
reference 20.
the two positive peaks, are evident in both the Kirch-
100
ho(cid:11) and FW{H solutions. These oscillations are al-
mostcertainlyduetoinaccuratequadratureoverpan-
els moving at high speed. The oscillations disappear
0
as the integration surface size is reduced. k=2
Now that the FW{H/RKIR code has been intro- k=7
duced, we wish to examine the sensitivity of each for- -100
k=12
mulation to the placement of the integration surface.
Brentneretal.5 foundthattheKirchho(cid:11)solutionvar- p¢, Pa k=18
ied somewhatwith locationof the integrationsurface. -200 k=21
Figure 2 shows a cross section of (cid:12)ve di(cid:11)erent inte- data
gration(Kirchho(cid:11))surfacelocationsrangingfromone
grid line o(cid:11) the surface to 1.37 chordlengths o(cid:11) the -300
surface. The Kirchho(cid:11) acoustic pressure predictions
from RKIR code for each of these surface locations
are shown in (cid:12)gure 3. As the integration surface is -400
00..00 00..55 11..00 11..55 22..00
brought nearer to the surface and the input data is
no longer compatible with the linear wave propaga- time, msec
tion assumption, the predicted acoustic pressure be-
comes meaningless. Although expected, this aspect of
the Kirchho(cid:11) method is troublesome. If the surface is
not positioned properly the error can be substantial. Figure 4. Comparison of predicted acoustic pressure
Worse yet, if the integration surface is just positioned using the FW{H formulation integration surface lo-
slightly in the nonlinear region the solution may be cations. These predictions are for an observer lo-
signi(cid:12)cantly in error but not enough so to be easily cated 3:4R from a UH-1H model rotor hovering at
recognized. MH = 0:88. The experimental data is from reference
20.
Figure4showsthenoisepredictionusingtheFW{H
6
111000000 flow direction
loading
mic 2
000
mic 6 mic 8
---111000000
30(cid:176) 30(cid:176)
ppp¢¢¢,,, PPPaaa
3.4 R
---222000000 thickness
FW-H/RKIR
WOPWOP+
---333000000 data
ww
total (includes quadrupole)
---444000000
000...000 000...555 111...000 111...555 222...000
tttiiimmmeee,,, mmmssseeeccc
Figure6. Schematicshowingthreeinplanemicrophone
locations used in the the measurement of noise from
Figure 5. Comparison of noise components predicted the model scale Operational Loads Survey (OLS) ro-
by the FW{H/RKIR and WOPWOP+ codes for a tor.21
hover UH-1H model rotor (MH = 0:88, inplane ob-
server 3:4R from rotor hub).
grationsurfacelocatedapproximately1.5chordlengths
away from the blade to predict the total noise. Note
formulation given in equation (17) for the same set thatthethicknessnoisepredictionsfromWOPWOP+
of integration surfaces and CFD input data as shown and FW{H/RKIR are identical and there is only a
in (cid:12)gure 3. The volume quadrupole source, which small di(cid:11)erence in the predicted loading noise. The
exists only outside the integration surface, has been di(cid:11)erence in the predicted loading noise is due to a
neglected in this calculation. The advantage of the di(cid:11)erence in how the integration over the blade tip
FW{H formulation is clear: for an integration surface faceishandled. Thetotalnoise,whichincludestheef-
near or on the physical body, the predicted acoustic fect of the quadrupole, is also in very close agreement
signalisessentiallythatofthicknessandloadingnoise even though the volume used in WOPWOP+ is not
alone. As the integrationsurfaceismovedfartherand identical to the region enclosed in the FW{H/RKIR
fartheraway,moreandmoreofthequadrupolesource surface integration. The negative peak is also in bet-
contribution is accounted for by the surface integrals. teragreementthantheearlier(cid:12)guresbecauseanEuler
17
Hence, we would say that the principal advantage of solution from Baeder was used as input rather than
the FW{H formulation for aeroacoustics is the relax- the FPRBVI solution used in (cid:12)gures 1, 3, and 4.
ation of integration surface placement restrictions. In A model-scale test of the Operational Loads Sur-
factwhenthevolumequadrupolesourceisincludedin vey (OLS) rotor is selected for a (cid:12)nal comparison.
the noise computation, the location of the integration The predicted noise from FW{H/RKIR, RKIR, and
surface is only a matter of choice and convenience. WOPWOP+ are compared with experimental data21
AnothertraditionaladvantageoftheFW{Hmethod atthreeinplanemicrophonepositions,shownschemat-
is the physical basis and identi(cid:12)cation of the source ically in (cid:12)gure 6. The rotor was operating in a for-
terms. Ifequation(17)isusedonasurfaceawayfrom ward(cid:13)ightconditionwithadvancing-tipMachnumber
thebodythisfeatureisnotretained,however,asecond MAT =0:84 and advance ratio (cid:22) =0:27. A FPRBVI
computationcanbemadeonthebodysurfacetodeter- solution (80(cid:2)36(cid:2)24 grid) was used as input data
mine thickness and loading noise. This has been done for all three noise predictions shown in (cid:12)gure 7. All
in (cid:12)gure 5, which is a comparison of FW{H/RKIR of these predictions agree quite well with the data|
predictionswithaWOPWOP+prediction. TwoFW{ bothindirectivityandamplitude. Allofthecodesun-
H/RKIR computations are show in (cid:12)gure 5: an inte- derpredict the negative peak pressure for microphone
grationsurfacecoincidentwiththerotorbladesurface 6, but this is most likely attributed to the FPRBVI
to predict thickness and loading noise, and an inte- solution rather that the noise prediction codes. The
7
40
0
Mic 2
(cid:252)
p¢, Pa -40 Mic 6 (cid:253) data
(cid:254)
Mic 8
FW-H/RKIR
-80
RKIR
WOPWOP+
-120
0 2 4 6 8 10 12
time, msec
21
Figure 7. Comparison of predicted and measured acoustic pressure at three microphone locations for the model
scale Operation Loads Survey (OLS) rotor (MAT =0:84; (cid:22)=0:27).
di(cid:11)erences between the predictions is most noticeable Kirchho(cid:11)methods. WeassumeallparametersofCFD
in the positive peaks, but even there predictions vary and acoustic calculations are dimensionless. Let V be
from each other by no more than 10 Pascals. the volume where CFD computations are performed
with the boundary @V. We de(cid:12)ne the Sobolev norm
A New Metric for Comparison of Tij as
ThequestionofwheretoplacetheKirchho(cid:11)surface,
(cid:26)ZTZ (cid:20)
athnedqthueadarnuaploolgeouinstqegureasttiioonn,ohfahvoewafastrroountgtiomppearcftoromn (cid:13)(cid:13)Tij(cid:13)(cid:13)V = XjTijj2+ X(cid:12)(cid:12)@@Txiij(cid:12)(cid:12)2
the decision of which method is most e(cid:14)cient com- 0 V i;j j
apcuctuartaiocnyaallny.d eTxwteontthoifngthsemCuFstDbceomcopnusitdaetrioend:nie)edthede + (cid:12)(cid:12) @2Tij (cid:12)(cid:12)2(cid:21)dxdt(cid:27)1=2
@xi@xj
as input data for noise prediction, and ii) the amount
(19)
of input data required. Both of these will depend on
the size of the nonlinear region surrounding the body
where T is a convenient time period usually taken as
generating the noise. While at present we are unable
the inverse of the blade passage frequency. We have
to give a completely satisfactory answer to the above
now a metric for comparing two CFD calculations as
questions,wecanprovidesomeguidelinesusinganew
follows. We de(cid:12)ne the distance (error) between two
metric as follows.
results by
We note that the solution of the FW{H equation
with the quadrupole source term invariably involves (cid:13) (cid:13)
Tij and its (cid:12)rst and second derivatives. Therefore, d(Ti1j;Ti2j)=(cid:13)Ti1j (cid:0)Ti2j(cid:13)V (20)
it is imperative that not only Tij is calculated accu-
1 2
rately, but also its (cid:12)rst and second derivatives in the where Tij and Tij pertain to the two sets of compu-
sourceregion. Similarly,theKirchho(cid:11)formulatellsus tational results. We may agree that the error is small
that on the Kirchho(cid:11) surface, p0, p_0, and @p0=@n(cid:17)p0n if (cid:13) (cid:13)
must be computed accurately in the CFD solution. (cid:13)Ti(cid:13)1j (cid:0)T(cid:13)i2j(cid:13)V (cid:28)1 : (21)
Thisindicatesthattheerroranalysisinallhighresolu- (cid:13)Ti1j(cid:13)V
tionCFDcomputationsmustbebasedontheSobolev
norm. This norm is used very often in (cid:12)nite element The two sets of results may come from two di(cid:11)erent
22
analysis and we propose such a norm in aeroacous- CFD computations.
tics. We will not present any numerical results in this Now we consider the Kirchho(cid:11) method. Assume S
paper based on the Sobolev norm. istheKirchho(cid:11)surfaceoverwhichthenondimensional
0 0 0
We(cid:12)rstaddresstheproblemofhowtocomparetwo p, p_ , and pn arespeci(cid:12)ed. We de(cid:12)ne aSobolev norm
0
high resolution CFD solutions for both FW{H and ofp anddistance for twosolutionsfrom CFD compu-
8
01 02
tations p and p as follows: types of wave propagation (e.g., the FW{H equation
is not appropriate for electromagnetic wave propaga-
(cid:26)ZTZ (cid:20) (cid:21) (cid:27)1=2
(cid:13) (cid:13) tion, while the Kirchho(cid:11) formula could be utilized).
(cid:13)p0(cid:13)S = jp0j2+ jp_0j2+jp0nj2 dSdt (22) ButthesuperiorityoftheFW{Hfortheaeroacoustics
0 S ofrotatingbladeshasbeendemonstratedthroughsev-
d(p01;p02)=(cid:13)(cid:13)p01(cid:0)p02(cid:13)(cid:13) : (23) eralnumericalexamples in this paper. Theplacement
S oftheintegrationsurfaceisamatterofconvenienceas
We can use this distance or error function to know long as the quadrupole source is utilized. The FW{H
when to stop a CFD grid re(cid:12)nement. Unfortunately, method also has the advantage that it separates the
this norm would not tell us when we are in the linear predicted noise into physical components (i.e., thick-
regionorwhetherthedispersionanddissipationerrors ness, loading, and quadrupole), explicitly. The Kirch-
0 0 0
have substantially in(cid:13)uenced p, p_ , and pn. These ef- ho(cid:11) method does not o(cid:11)er this insight intothe nature
fects are governedby grid size as well as arti(cid:12)cial vis- of the acoustic (cid:12)eld.
cosity. In the study of these e(cid:11)ects in high resolution It is well known that the quadrupole sources are
CFD calculations, we must employ a Sobolev norm in responsible for noise generation as well as distortion
de(cid:12)ning the computational errors. of the acoustic waveform. The intense quadrupole
An alternate use of the norm de(cid:12)ned in equa- sources are in the vicinity of the blades. Therefore,
tion (19)isto decidethe volumeof quadrupolesource if we use a surface which encloses the blade and the
included in our noise calculations. Let V1 and V2 be volume of intense quadrupoles in the FW{H method,
two volumes such that V1 (cid:26) V2. Then assume that we can calculate the level of the acoustic pressure ac-
1
Tij =0 outside V1. Using the Sobolev norm with vol- curately. The role of the weaker quadrupoles, which
ume integration over V2, we can say that V1 includes are farther away from the physical body, is primarily
all quadrupoles needed for noise calculation if toprovideasmalldistortiontotheacousticwaveform.
(cid:13)(cid:13)Ti(cid:13)1j (cid:0)T(cid:13)i2j(cid:13)(cid:13)V2 (cid:28)1 : (24) Htoenthcee,neoviesnewgehneenrathtienigntseugrrfaatcieo,nitsumrfaaycebiesfaacicrelyptcalobslee
(cid:13)Ti1j(cid:13)V1 to neglect the external quadrupoles. In comparison,
the Kirchho(cid:11) formula can predict acoustic pressures
This means that
that are substantially in error if the Kirchho(cid:11) surface
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13)Ti2j(cid:13)V2nV1 (cid:28)(cid:13)Ti1j(cid:13)V1 (25) is locatedinside the nonlinearregion. The natureand
order of magnitude of this error may be hard to esti-
whereV2nV1 is the volumeenclosedbetween@V1 and mate or even recognize.
@V2. This answers how far from the blade surface we
must include quadrupole sources. References
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