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Kenneth S. Brentner?

NASA Langley Research Center
Hampton, Virginia


A new formulation for the Ffowcs Williams{ Hawkings quadrupole source, which is valid for a far-field in-plane observer, is presented. The far-field approximation is new and unique in that no further approximation of the quadrupole source strength is made and integrands with r?2 and r?3 dependence are retained. This paper focuses on the development of a retarded-time formulation in which time derivatives are analytically taken inside the integrals to avoid unnecessary computational work when the observer moves with the rotor. The new quadrupole formulation is similar to Farassat's thickness and loading formulation 1A. Quadrupole noise prediction is carried out in two parts: a preprocessing stage in which the previously computed flow field is integrated in the direction normal to the rotor disk, and a noise computation stage in which quadrupole surface integrals are evaluated for a particular observer position. Preliminary predictions for hover and forward flight agree well with experimental data. The method is robust and requires computer resources comparable to thickness and loading noise prediction.

2 wave operator
c sound speed in undisturbed medium d? element of the collapsing-sphere surface dS element of the rotor-blade surface f = function that describes the rotor-blade
f+ = surface described by union of rotor-blade
surface and rotor disk

?Research Engineer, Aerodynamic and Acoustics Methods Branch, Fluid Mechanics and Acoustics Division, Senior Member AIAA.

Copyright c 1996 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for government purposes. All other rights are reserved by the copyright owner.

g = surface that describes the collapsing sphere, g = o ? t + r=c
H(f) Heaviside function
li local force that acts on body
~M local velocity vector of source normalized by c, with components Mi
_Mi @Mi=@o
Mr Mach number of source in radiation direction, Mi^ri
_Mr _Mi ^ri
?Mr ?Mi ^ri
MAT advancing tip Mach number
MH hover tip Mach number
^n unit outward normal vector to surface, with components ^ni
p0 acoustic pressure, p ? po
Qij quadrupole surface source tensor, symmetric QMM QijMiMj
QMr QijMi^rj
Q _M r Qij _Mi^rj
Qrr Qij^ri^rj
_QMr _Qij Mi^rj
_Qrr _Qij ^ri^rj
?Qrr ?Qij ^ri^rj
r distance between observer and source, r = j~x ? ~yj
^r unit vector in the radiation direction, with components ^ri
R rotor radii
t observer time
Tij Lighthill stress tensor, aeuiuj + (p0 ? c2oae0)ffiij (inviscid form)
vn local normal velocity of source surface ~x observer position vector, with components xi ~y source position vector, with components yi

Greek symbols:
? intersection of collapsing sphere g = 0, and

source surface f = ffi (f) Dirac delta function
? angle between ^n and ^r
aeo density of undisturbed medium