Second Computational Aeroacoustics (CAA) Workshop on ...

_:C-_ and _=,.o/ct32.
Equation (8) casts the boundary integral equation in a form from which its singular behavior may
be extracted explicitly. The analysis necessary to do so is fairly lengthy, however, and it will be only
summarized here in symbolic form; more detail will be found in a forthcoming publication [7]. First, it
is seen in eq. (9) that the singularity occurs when Y3=X3(_=0),_=0 and R-a-h=0. Expansion of
for small _,_,h results in
1 i_- R*2
(_Z+Bo)3/z j (_2 +B0)3/2
in which Bo=[32(h2+a2_ 2) and the omitted terms are all O(1) or smaller. Let the terms shown in eq.
(10) be denoted as _o(Y3,#) and define q=_-_0- This function is completely regular. Then the
circumferential integral in eq. (8) is written as
f@d* +fVod* =I._(V_)+I(Y_)
0 o
such that the first integral (Ins) is nonsingular and the second (I) can be evaluated analytically because it
contains only quadratic expressions in _. After carrying out this integration it is found that
I(Y3) = Q,_(Y3) ÷ Qs(Y3), where Q,_ is also completely nonsingular at Y3 =X3, h=O. The term Qs
contains the singularity and is discussed further below.
If the results just described are substituted back into eq. (8), that equation becomes
-4r_P(X3) :2a[32 I? _(Y3)
l-L/2 -ff_ jR= dY3 + lima._ aRj
after the radial derivative of the nonsingular integrals on R=a has been evaluated analytically, and in
which
-u2[ ¢_2 +[32Cn2+aR_2) J_2+[32h2
The entire singular nature of eq. (18) is now isolated in the integral (13) and the final step in the analysis
is to remove the singularity from this integral by appropriate expansion about _ =0 of the nonsingular
bracketed factor in the integrand of (13). When this is carried out it is found that the last term on the
right of eq. (12) is expressible as
22
+
• , ,
=:
(lO)
(11)
(12)
(13)
=
[
l