THESE de DOCTORAT - cerfacs

44 Chapter 3: Development of a numerical tool for combustion noise analysis, AVSP-f
n
e k+α1
n
N i
e k+α2
e k+α3
e k+α4
e k+α5
e kn
n
n
n
Figure 3.2: A set of six triangular cells embedded in a mesh grid
S 1,i
e k+α1 n
e k
n
n
e k+α2
N i
S 2,i
e j+4
⃗n
e j+3
n n
⃗n
e j+5
Figure 3.3: A set of three triangular cells at the boundary of the computational domain
⃗n
to the computational boundary.
∮
1
¯c 2 N
V i
∇ ˆp
Ni
⏐ · ndS = 1
∂N i ek+α
( ∮ )
¯c 2 N
V i
h ek+α ( ˆp o ) · ndS
Ni S 1,i
+ 1 ( ∮ )
¯c 2 N
V i
h ek+α ( ˆp o ) · ndS
Ni S 2,i
(3.6)
The flux of the pressure gradient across the surface S 2,i is associated to the boundary conditions
imposed to the surface. This boundary condition can be expressed in terms of an acoustic
impedance Ẑ and pressure ˆp. Hence (∇ ˆp · n S2,i ) = f ( Ẑ Ni , ˆp i ) ) . Expression (3.6) becomes
∮
1
¯c 2 N
V i
∇ ˆp
Ni
⏐ · ndS = 1
∂N i ek+α
( ∮ )
¯c 2 N
V i
h ek+α ( ˆp o ) · ndS
Ni S 1,i
+ 1 ( ∮ )
¯c 2 N
V i
f (Ẑ Ni , ˆp i )dS
Ni S 2,i
(3.7)
Finally, the Helmholtz equation Eq. (2.63) for the node N i reads