Second Computational Aeroacoustics (CAA) Workshop on ...

I. ANALYTICAL SOLUTION OF THE PROBLEM 2
This is an initial-value problem governedby the linearized Euler equations
The initial conditions are:
and
Here xs = 4, w = 0.2.
and
by
The boundary conditions are:
Ou Op
-_ + N = 0 (28)
Ov Op
N + _yy = 0 (29)
Op Ou Ov
N + _ + N = 0 (30)
u=v=0 at t 0 (31)
p(x, y, 0) = e -u'2((,-*'):+y2)/w2
l
at t = 0 (32) _*
v.n=O at x 2+y2=(0.5) 2, (33)
when x, y ---+ oc the solution respresents outgoing waves. (34)
Solution of the problem (28) - (34) can be found in terms of velocity potential ¢(x, y, t) defined
0¢ 0¢ 0¢
= N' "= 0_' p - ot (35)
It is easy to show from (28) - (30) and (35) that the governing equation for ¢ is the wave equa-
tion which may be written in polar coordinates (r, O) as
Initial conditions (31) and (32) are reduced to
t=O:
1
02¢ f82¢ 10¢ __02¢_
Ot 2 \ Or 2 + r -_r + r 2 002 ,] = 0 (36)
¢=0, Ot - e (37)
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