THESE de DOCTORAT - cerfacs

46 Chapter 3: Development of a numerical tool for combustion noise analysis, AVSP-f
j
j
j
i
i
i
a) A i,j b) B i,j c) C j C T j
Figure 3.4: Typical Matrices for a non-structured 3D problem [93]. Non-zero entries are shown
as dark points. Here C T is the transpose of C. The dyadic product CC T is shown to display the
non-zero elements of C j
3.2 Boundary Conditions in AVSP-f
There are three types of acoustic boundaries in the numerical tool AVSP-f. The first one is of
the Dirichlet type. It means that at the boundary a zero pressure fluctuation ˆp is imposed. This
represents usually the boundary condition used at the outlet of the domain when this one is
open to the atmosphere. In order to impose ˆp = 0, it is necessary to remove the concerned
nodes from the matrix A i,j so that the linear system to resolve is well-posed. Equation (3.9)
becomes:
[
Ai,j + B i,j
] ˆpo = C j ⇒
(A i,j
⏐
⏐⏐⏐i,j̸∈∂SD
)
ˆp o = C j (3.13)
where S D stands for the surfaces in which the homogeneous Dirichlet boundary condition is
applied. Another important boundary condition corresponds to totally reflecting boundaries,
usually applied to walls and inlets. It is of the type of Neumann since what is imposed here
is the gradient of ˆp. The linearized momentum equation for low Mach number flows in the
frequency domain reads
iω ¯ρû · n = ∇ ˆp · n (3.14)
It is clear that no fluctuations of velocity are allowed normal to the surface (û · n = 0) if the
gradient of the fluctuating pressure normal to the boundary is zero (∇ ˆp · n = 0). For this
homogeneous Neumann boundary condition, the system to resolve is