Second Computational Aeroacoustics (CAA) Workshop on ...

Substitution of (46)into (44), (45) gives
(47) can be solved by separation of variables giving,
02At 1 OAr 10ZA,.
Or 2 + + +ofiA = 0 (47) :
r Or r 2 002 ._
_rr = B(6,o_) at r =0.5 (48)
|
where B(0,0J)= OA.___i ]
(49) i
Or It=0.5
O0
A4r, e,_) = _ Ck(_)lC_l)(r_)co_(ke) (50)
k=0
The coefficients Ck of (50) are found by applying the boundary condition (48). It is straight-
forward to find,
_ B(e,_)cos(ke)de (51)
Ck = 7rw[ZkH(1) ' ,,,,
--d k (w/z)- g_l(w/2)]
Finaiiy, substitution of (50), (51) into (46) and on combining (42), (46), the velocity potential
is found,
where
¢(r,e,t) = A(r,e,_)_i.(_t)d_ (52)
j00 _
A(_,e,_) = Jo(_/_ _+ _ - 2_z._o.e) + F_.n_
g-bw=/(4b){2/) OO [ 71"_3[_-H_I) (O2/2)_kH(kl)(Fc'O)g°'S(ICe)-k=0
.. D "(1) k+l (w/2)]
r + (53)
]0 v/0.25 + x] - x,cos_ J
The pressure field may be calculated by
p(_,e,t) = _o¢ = _ A(_,e,_)_¢o.(_t)d_
Ot
f0 _
REFERENCES
1. Abramowitz, M.; and Stegun, I.A.: Handbook of Mathematical Functions.
2. Tam, C.K.W.; and Webb, J.C.: Dispersion-Relation-Preserving Finite Difference Schemes for
<strong>Computational</strong> Acoustics. J. Comput. Phys., vol. 107, Aug. 1993, pp. 262-281.
14
(54)
=
I
!
i[]
m