NASA Technical Reports Server (NTRS) 20040086848: Verification and Analysis of Formulation 4 of Langley for the Study of Noise From High Speed Surfaces

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AIAA 99-1881
Verification and Analysis of Formulation 4
of Langley for the Study of Noise From High
Speed Surfaces
F. Farassat
NASA Langley Research Center
Hampton, VA
Mark Farris
Midwestern State University
Wichita Falls, TX
5th AIAA/CEAS AeroacousticsConference
10–12 May 1999
Bellevue (Greater Seattle), WA
For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics
1801 Alexander Bell Drive, Suite 500, Reston, VA 20191
AIAA-99-1881
Verification and Analysis of Formulation 4 of Langley for the Study of Noise From
High Speed Surfaces
F. Farassat
AIAA Associate Fellow, Senior Research Scientist
NASA Langley Research Center, Hampton, Virginia
Mark Farris
Midwestern State University, Wichita Falls, Texas
Abstract execution time on a computer is long compared to the
subsonic formulations. There are many surface geomet-
There are several approaches to the prediction of
ric parameters, such as local normal curvature in various
the noise from sources on high speed surfaces. Two of
directions, in Formulation 3 which can not be physically
these are the Kirchhoff and the Ffowcs williams-Hawk-
interpreted. It is difficult to assess the accuracy of noise
ings methods. It can be shown that both of these meth-
prediction because of the complexity of the computing
ods depend on the solution of the wave equation with
algorithm. We have searched for simpler results for pre-
mathematically similar inhomogeneous source terms.
diction of noise from sources on high speed surfaces.
Two subsonic solutions known as Formulation 1 and 1A
of Langley are simple and efficient for noise prediction. Farassat, Dunn and Brentner have presented a new
The supersonic solution known as Formulation 3 is very result in the last AIAA Aeroacoustics Conference in
complicated and difficult to code. Because of the com- Toulouse which is considerably simpler than
plexity of the result, the computation time is longer than Formulation 37. This result has been designated
Formulation 4. The present paper continues the study of
the subsonic formulas. Furthermore, it is difficult to
this new result. We apply Formulation 4 to two problems
assess the accuracy of noise prediction. We have been
whose analytic solutions are known by other methods.
searching for a new and simpler supersonic formulation
These are: i) the noise from dipole distribution on the
without these shortcomings. In the last AIAA Aeroa-
unit circle whose strength varies radially with the square
coustics Conference in Toulouse, Farassat, Dunn and
of the distance from the center and ii) the noise from
Brentner presented a paper in which such a result was
dipole distribution on the unit sphere whose strength
presented and called Formulation 4 of Langley. In this
varies with the cosine of the angle from the polar axis.
paper we will present two analytic tests of the validity
We show that we do obtain the known analytic results
this Formulation: i) the noise from dipole distribution on
and thus have validated Formulation 4.
the unit circle whose strength varies radially with the
square of the distance from the center and ii) the noise We discuss the question of singularities of the new
from dipole distribution on the unit sphere whose formulation which surprisingly is simpler to answer
strength varies with the cosine of the angle from the than those of Formulation 3. We was shown that the sin-
polar axis. We will discuss the question of singularities gularities are removable for FW-H equation if we
of Formulation 4. include the surface terms from the quadrupole source,
and in the Kirchhoff formula for supersonic surfaces.
Introduction
The Governing Equation and Its Solution
Two common methods of noise prediction from
moving surfaces are based on the Ffowcs Williams- Given an open moving surface f = 0, ˜f >0,
Hawkings (FW-H) equation1 and the Kirchhoff formula wheref = ˜f = 0 denotes the edge of the panel, it can
for moving surfaces2. It can be shown that both these be shown that the governing differential equation for
methods are based on the solution of wave equation with noise prediction by FW-H equation and the Kirchhoff
mathematically similar inhomogeneous source terms. method is7:
The subsonic solutions known as Formulations 1 and 1A
of Langley3,4,5 are simple and efficient to use on a com- 2p¢ = q H(˜f) d (f)+q H(f)d¢ (f)
puter. The supersonic result known as Formulation 3 is 1 2 s
˜ (1)
very complicated and difficult to code for noise +q d (˜f) d (f)
3
prediction6. Because of the complexity of this result, the
1
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where the functions q ,q and q are described in and Formulation 4 is valid for both subsonic and super-
1 2 3
reference 7. In this equat˜ion, H(.) is the Heaviside sonic surface sources. We consider two problems here.
function, d (.) is the Dirac delta function and d¢ (f) is
s
a distribution that picks up normal derivative of a test Example 1- Dipole Distribution on the Unit Circle
function on the surface f = 0. The full solution of this
We consider dipole distribution on the unit circle
equation (Formulation 4) is7:
with the center at the origin of the x x -plane described
1 2
by the following wave equation:
4p p¢ (x,t)
= (cid:242) 1--- q----1----+-----c--o---t---q----t---1----(cid:215)---(cid:209)----2--q---2---–--- --k----1---q---2- dS 2p¢ = ---¶-----[q(x ,x ,t)d (x )]
r L ret ¶ x3 1 2 3 (3-a)
F = 0
F˜ >0 = q(x ,x ,t)d¢ (x )
q +q n (cid:215) t cotq (2) 1 2 3
+(cid:242) 1--- ----3------------2-------------1------------- dL
F = 0 r L 0 ret q = –(1+r 2)eiw t (3-b)
F = 0 2
+S r-----1--4--–-q----2M-----r---(cid:242) 0p ⁄2s-k--i-r-g---–-[--k-k--(--(j--j---)-)-]-dj T r 2 = x12+x22, r £ 1 (3-c)
Here, r = x –y , (x,t) and (y,t ) are the observer The solution of this problem from classical mathematics
and the source space-time variables, respectively, and q is
is the angle between the radiation direction rˆ = r ⁄r
and the local normal to f = 0. The unit vector in the 4p p¢ (x,t) =
direction of projection of rˆ on the local tangent plane to
the source surface is denoted t1 and the local normal x eiw t(cid:242) 1(cid:242) 2p e---–---i--k--r-(1+r 2)(1+ikr)r jd dr (4)
curvature of f = 0 in the direction of t1 is k 1. the 3 0 0 r3
geodesic unit normal of the edge of the panel is n and
L and L 0 are functions of the kinematic and geometric where (r j, ) is the polar coordinates in the x x -plane,
1 2
parameters of the panel7. We have defined k = w ⁄c and r is the distance between the source and
F = [f(y,t )] and F˜ = [˜f(y,t )] . The last
ret ret the observer. We will later integrate Eq. (4) numerically
term only exists if the collapsing sphere
to compare with the results from Formulation 4.
g= t –t–r⁄c = 0 leaves the panel tangentially at the
point T. The signum function is denoted sig(.), Now we use Formulation 4, Eq. (2), for solving
k = 1⁄r and k(j ) is the local normal curvature of the Eq. (3). Refer to Fig. 1 for definition of some symbols.
r
panel at T as a function of azimuthal angle j . The Mach Because of the symmetry of the problem with respect to
number in the radiation direction is Mr. We mention the x3-axis, we assume that the observer is in the x1x3-
here that Formulation 4 is valid at all Mach numbers plane. We have the following relations:
although we intend to use it for surfaces moving at tran-
sonic and supersonic Mach numbers. r2 = r 2+x2+x2–2r x cosj (5-a)
1 3 1
Note that we have issued a correction to the result
presented in reference 7. The correction appears in the
r2 = r 2+x2–2r x cosj (5-b)
electronic copy of this reference at NASA Langley 1 1 1
Technical Report Server. The electronic address is given
in the references below. cosq = x ⁄r, sinq = r ⁄r, cotq = x ⁄r (5-c,d,e)
3 1 3 1
Validation of Formulation 4
Since the part of the new formulation depending on t1 = Ł(cid:230) -x---1---–-----rr----c---o--j-s-----,-–---r---r-s--i--nj------,0ł(cid:246) (5-f)
q and q are simple and have been validated before6, 1 1
1 3
we only need to validate the part depending on the
source term q2. We will again start with the differential 2r eiw t(x cosj –r )
equation and assume that the sources are stationary. It t (cid:215) (cid:209) q = – -----------------------1-------------------------- (5-g)
1 2 r
will be seen that these assumptions are necessary 1
because we are seeking problems with analytic solutions
2
American Institutes of Aeronautics and Astronautics
k 1 = 0, k g = (1⁄r1), (5-h,i) Now, for the observer on the x3-axis and in the far
field, Formulation 4, Eq. (6) gives:
n (cid:215) t = (1–x cosj )⁄r (5-j)
1 1 1 1
p¢ (x,t) = ------------(cid:242) [cotq t (cid:215) (cid:209) q ] dS
4p r 1 2 2 ret
0
Using these results in Eq. (2), we get
(cid:230) k (cid:246)
p¢ (x,t) = – -x---3-2-e--p-i--w----t(cid:242) 01(cid:242) 02p r----2---e---–--i--k--r---(--x-r--1-r--c12---o---s---j-----–-r-------)-dj dr –– 4---r---p--1--q--r-----2--0--(cid:242) Ł(cid:231) s---i-n----12--q---+k –gcor---t-q--qł(cid:247)-2-- q2 retdS (9)
+ -x---3--e---i--w----t(cid:242) 2p -x---1---c--o---s---j-----–-----1--e–ikr dj 2r0 t–(r0+1)⁄c 2r0 t–(r0–1)⁄c
2p 0 r2 Here, q2 is the right side of Eq. (7) and dS is element
1 r =1 of the surface area of the sphere . We next use the fol-
iw t lowing results in Eq. (9):
+ e---------(1+x2)e–ikx3H[1– x2 ] (6)
2 1 1
q =y , k = –1, r = 1, k = coty (10-a,b,c,d)
1 g
The two expressions in Eq. (4) and Eq. (6) look very dif-
fceormenptu ftreo mp¢ (exac,ht )oet–hiew rt. Wfreo mha vthee sues etdw Mo aetxhpermesastiiocnas 3 f otor t1(cid:215) (cid:209) 2q2 = – ¶-y--¶----–[cosy eiwt ]
(11)
11 values of x1. In these calculations, shown in Table 1, = –siny eiwt
we used k = 10, x = 0 and x = 5. It is seen that
2 3
the results from the two expressions are the same to a
k
remarkable degree of accuracy.
--------1-----+k cotq = –1 (12)
This example validates Formulation 4 for a flat sin2q g
source surface. The next example applies this result to a
curved surface. r q
2
– ---------
Example 2- Dipole Distribution on a Sphere 2r0 t–(r0+1)⁄c
(13)
We will consider a unit sphere R = 1 with the cen- – r----q---2-- = i---s--i--n---k--ei(w t–kr0)
ter at the origin and a dipole distribution varying with 2r0 t–(r0–1)⁄c r0
the cosine of the angle Y from the x -axis. See Fig. 2
3
for some notation. We consider the following wave In Eq. (11), the symbolt stands for the source time that
equation: can be related to the angle y on the surface of the
sphere. When we use the above results in eq. (9), we get
2p¢ = – eiw tcosY d¢ (R–1) (7) exactly the classical results Eq. (8). We have thus vali-
dated Formulation 4 for a curve surface also.
We use (R,Y F, ) and (r y,j , ) for the observer and
Discussion of the Singularities
the source variables, respectively. Let r be the distance
0
of the observer from the origin. Then, the solution of Eq. One of the problems associated with supersonic
(7) in the geometric far field when the observer is on the surface sources is the appearance of singularities in the
positive x -axis is: solution of wave equation. Some of these problems are
3
purely mathematical in nature and their cause is the
i sink i(w t–kr ) wrong choice of variables in the solution of the wave
p¢ (x,t) = -------------- e 0 (8)
r equation. There is also the possibility of physical singu-
0
3
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larities where no choice of variables can get rid of. We a sphere. To get an analytically simple expression from
mention that both the thickness and loading sources on classical analysis, the observer is located in the far field
an open supersonic surface will have true singularities at and on x -axis. We showed that this result could also be
3
some observer time. This problem was treated by Di obtained by the new formulation.
Bernardis8 and Farassat and Myers9. The latter authors
The most significant fact about the new formulation
showed that the inclusion of surface sources from the
is that it is much simpler than any previously known
quadrupole source term of FW-H equation in the solu-
result in time domain for prediction of the noise from
tion of this equation results in integrable singularities.
high speed surface sources. Furtheremore, because of
We have shown that similar conclusion holds for the
the observer location, in the case of propfan noise calcu-
new formulation when applied to the solution of FW-H
lations, none of the problems of singularities are
equation and the governing equation for the Kirchhoff
present. This appears to be a major advance in noise pre-
formula for moving surfaces7.
diction theory.
The singularities of Formulation 3 for an open
supersonic surface appear when part of its edge travels Referensce
at supersonic speed in the plane normal to the edge. One
1. Ffowcs Williams, J. E. and Hawkings, D.L., “Sound
can then construct the observer positions and the times
generation by turbulence and surfaces in arbitrary
that the singularity will be felt at the observer. The situa- motion”, Phil Trans. Roy. Soc. (London), 264A ,
tion for Formulation 4 is somewhat different. First the 321–342 (1969)
singularities from the surface and line integrals are 2. Farassat, F., “The Kirchhoff formulas for moving
surfaces in aeroacoustics—The subsonic and
much simpler to analyze than those of Formulation3 but
supersonic cases”, NASA Technical Memorandum
of the same nature. Another cause of the appeance of
110285 (1996), (Available at ftp://
singularities is due to the geometry of the source surface techreports.larc.nasa.gov/pub/techreports/larc/96/
itself and is related to the formation of the caustic in NASA-96-tm110285.ps.Z)
geometric acoustics10. This type of singularity comes 3. Farassat, F., “Theory of noise generation from mov-
ing bodies with an application to helicopter rotors”,
from the last term of Eq. (2). We will discuss the prob-
NASA TR R-451 (1975), (Available at http://techre-
lem of singularities in a comprehensive paper on the
ports.larc.nasa.gov/ltrs/PDF/NASA-75-trr451.pdf)
new formulation later. 4. Farassat F. and Succi, G. P., “The prediction of heli-
copter rotor discrete frequency noise”, Vertica, 7,
Concluding Remarks 309–320 (1983)
5. Brentner, Kenneth S., “Prediction of helicopter
The purpose of this paper has been to validate For- rotor discrete frequency noise—A computer pro-
mulation 4 of Langley for prediction of noise from high gram incorporating realistic blade motions and
advanced acoustic formulation”, NASA Technical
speed moving surfaces. We have used two problems for
Memorandum 87721 (1986)
which analytical solutions are available from classical
6. Farassat, F., Padula, S. L. and Dunn, M. H.,
analysis. We have shown that these solutions can be “Advanced turboprop noise prediction based on
obtained also using the new formulation. The first prob- recent theoretical results”, J. of Sound and Vib.,
lem is the radiation field of dipole distribution on a flat 119, 53–79 (1987)
7. Farassat, F., Brentner, Kenneth S. and Dunn, M. H.,
surface. We verified by a numerical study that the radia-
“A study of supersonic surface sources—The
tion field can be obtained by the new formulation. The
Ffowcs Williams-Hawkings equation and the
second problem is radiation from dipole distribution on Kirchhoff formula”, AIAA Paper 98-2375 (1998)
4
American Institutes of Aeronautics and Astronautics
(Available at http://techreports.larc.nasa.gov/ltrs/
Observer (x , 0, x )
PDF/1998/aiaa/NASA-aiaa-98-237 x
1 3
8. Di Bernardis, Enrico: On a New Formulation for 3
the Aeroacoustics of Rotating Blades (in Italian),
Ph.D. Thesis, University of Romr (La Sapienza),
1989 r
9. Farassat, F. and Myers, M. K.: Line Source Sin-
gularity in the Wave equation and Its Removal by x
Quadrupole sources – a Supersonic Propeller Noise 2
Problem, in Theoretical and Computational Acous- q
tics, Volume 1, J. E. Ffowcs Williams, D. Lee, and
n
A. D. Pierce (eds.), World Scientific Publishing,
t r
1994
r 1
10. Pierce, Allan D.: Acoustics – An Introduction j x
1
to Its Physical Principles and Applications, Acous-
tical Society of America, 1989
Figure 1. Definition of some symbols in Example 1
x
3
Source
r
r
Q
Observer
R
x
x 2
1
Figure 2. Definition of some symbols in Example 2
5
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Table 1. Numerical Comparpi¢ (sxo,nt) eo–ifw t From Eq. (4) and Formulation 4, Eq.
k = 10, x = 0 x, = 5 .
2 3
Formulation 4
x Classical, Eq. (4)
1 Eq. (6)
0 0.21098 + 0.673096 i 0.21098 + 0.673096 i
0.25 0.23068 + 0.637725 i 0.238067 + 0.637724 i
0.50 0.30169 + 0.531172 i 0.301689 + 0.531171 i
0.75 0.355626 + 0.361800 i 0.355625 + 0.3618 i
0.975 0.353059 + 0.184642 i 0.353058 + 0.184641 i
0.995 0.34963 + 0.169323 i 0.349588 + 0.169312 i
0.9995 0.348784 + 0.165908 i 0.348784 + 0.165908 i
0.99995 0.348698 + .165567 i 0.348698 + 0.165567 i
1.00005 0.348679 + 0.165491 i 0.348679 + 0.165491 i
1.25 0.263827 + 0.006876 i 0.263827 + 0.006876 i
5 0.00397175 - 0.0118504 i 0.00397175 + 0.0118504 i
6
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VERIFICATION AND ANALYSIS OF
FORMULATION 4 OF LANGLEY FOR THE
STUDY OF NOISE FROM HIGH SPEED
SURFACES
F. Farassat
NASA Langley Research Center, Hampton, Virginia
Mark Farris
Midwestern State University, Wichita Falls, Texas
Presented at
The 5th AIAA/CEAS Aeroacoustics Conference
Seattle, WA, 10-12 May 1999
NNAASSAA LLaanngglleeyy RReesseeaarrcchh CCeenntteerr 1 of 13
F. Farassat and Mark Farris
Outline of the Talk
– A short history of development of time
domain formulations at Langley
– The governing wave equation
– Formulation 4
– Analytic validation by two examples
– The issue of singularities
– Concluding remarks
F. Farassat and Mark Farris 2 of 13
A Short History of Development of
Time Domain Formulations at Langley
We have been interested in helicopter rotor and
propeller noise prediction since early 70’s. We
*
have developed :
i) Formulations 1 and 1A for subsonic surface
motion with Doppler factor- highly efficient for
noise prediction
ii) Formulation 3 for subsonic, transonic and
supersonic surface motion without the Doppler
factor- very complicated, difficult to code and
inefficient for noise prediction
We need a replacement for Formulation 3!
* Formulation 2 was abandoned because of its limitations
F. Farassat and Mark Farris 3 of 13