THESE de DOCTORAT - cerfacs

3.1 Discretizing the Phillips’ equation 45
[ ∇
( ¯c 2 ∇ ˆp ) + ω 2 ˆp ] N i
= −iω(γ − 1) ˆ˙ω T
⏐ ⏐⏐⏐Ni
( ∮ )
1
¯c 2 N
V i
h ek+α ( ˆp o ) · ndS + 1 ( ∮ ) (3.8)
¯c 2 N
Ni S 1,i
V i
f (Z Ni , ˆp i ) dS + ω ˆp i = −iω(γ − 1) ˆ˙ω T,i
Ni S 2,i
Equation (3.8) must be solved for each node N i of the computational domain. In matrix representation,
this equation can be written as
[
Ai,j + B i,j
] ˆpo = C j (3.9)
where
A i,j ˆp o = 1 ( ∮ )
¯c 2 N
V i
h( ˆp o ) · ndS + ω 2 ˆp i (3.10)
Ni S 1,i
B i,j ˆp o = 1 ( ∮ )
f (Z Ni , ˆp i )dS
V Ni S 2,i
(3.11)
C j = −iω(γ − 1) ˆ˙ω T,i (3.12)
The matrix A i,j is a n × n sparse polydiagonal matrix where n stands for the total number
of nodes in the computational grid. The number of diagonals depends on the degree of the
interpolation used to relate the neighboring values of N i to obtain the respective value at N i .
It should be clear then that A i,j ̸= 0 when A i,j = A i,o , or in other words, A ij is non-zero for
the neighboring nodes implied into the respective interpolation procedure. As a result A i,j = 0
if j /∈ o, i.e., j > i + l, and if j < i − l. As a consequence, A i,j = 0 for elements far from the
mean diagonal. This can be observed in Fig. 3.4a. Let us recall that l can contain high values
for unstructured meshes.
Matrix B i,j is a n × n sparse polydiagonal matrix and is associated to the boundary conditions.
It is non-zero when both the node N i lies on one of the boundary surfaces and Ẑ Ni ̸= 0. As a
consequence, B does not play any role when Neumann boundary conditions are used: Ẑ Ni = 0
or û ∝ ∇ ˆp = 0, i.e., when solid boundaries (walls) are considered. A typical distribution of the
non-zero elements of matrix B i,j is shown in Fig. (3.4b)
C i,j is a vector of size n. It contains the sources of the acoustic problem, in our case the unsteady
heat release rate ( ˆ˙ω T ). The majority of its values are zero, except for those ones where the flame
is present. The matrices A i,j , B i,j and the vector C j are shown in Fig. (3.4)