Radiation Boundary Conditions

Подпись: p pa й ua V V a p pa Подпись: + Or 2 Подпись: (6.1)

At boundaries where there are only outgoing acoustic waves, the solution is given by an asymptotic solution (5.25) as follows:

where V(e) = a0[M cos в + (1 – M2 sin2 в)1/2]. The subscript ‘a’ in (pa, ua, va, pa) in Eq. (6.1) indicates that the disturbances are associated with the acoustic waves alone. By taking the time t and r derivatives of formula (6.1), it is straightforward to find that for arbitrary function F the acoustic disturbances satisfy the following equations:

P

Подпись: (6.2)Подпись: 0 + O/ 1 д д й

V (в) дt дГ 2r/ V

p

Eq. (6.2) provides a set of radiation boundary conditions.

where the explicit form of (pa, ua, va, pa) may be found in Eq. (6.1). The functions X, f, and F are entirely arbitrary. It is observed that the total pressure fluctuation comes only from the acoustic component of the outgoing disturbances. Thus, the appropriate outflow boundary condition for p is the same as that of Eq. (6.2). On writing out Cartesian coordinates in two dimensions, it is

 

d p.9 p p

+ cos в — + sin в — + ^~ d x dy 2r

 

(6.4)

 

0,

 

where в is the angular coordinate of the boundary point.

By differentiating the expression for p in Eq. (6.3) with respect to t and x, the following equation is found:

 

dp dp dp dp„

— + U0—— = — + u0—. 91 0 dx 91 0 dx

 

(6.5)

 

But pa = pa/a2 = p/a0 andp is known from Eq. (6.4). By eliminating pa, the outflow boundary condition for p becomes

 

du 9 u 1 9 p 91 0 dx p0 dx

9 v 9 v 1 9 p

dt ^ U° 9x p0 9y ’

In summary, the outflow boundary conditions are as follows:

 

(6.11)

(6.12)

 

(6.13)

 

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