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106 Chapter 6: Boundary conditions for low Mach number acoustic codes
to be taken specially with the use of the inlet boundary condition, since it must impose
both the mean flow and either the entropy or acoustic wave [37]. Therefore, the inlet
must be non reflecting so that the reflected w − 1
be completely evacuated. The mesh is
composed of sixty thousand nodes which is equivalent to the two thousand nodes mesh
for the Quasi-1D LEE solver.
6.4.2 Results
The value of both transmission and reflection coefficients are shown in Fig. (6.4). They are
plotted against the non-dimensional frequency Ω = ωl n / ¯c 1 where ω is the angular frequency,
l n the length of the nozzle and ¯c 1 the speed of sound at the inlet. It is observed that analytical
results are well suited when considering small frequencies (Ω → 0). Both numerical tools
converge to Marble & Candel expressions when λ ≫ l n for all coefficients, with the exception
of R SA where the AVBP solver, which solves the 3D Euler equations, fails to reproduce the
analytical value. Despite our efforts, no definite explanation was found for this result, even if
artificial viscosity and lack of grid refinements seem to be good candidates for explaining such
errors. Note however that the error is for the R SA coefficient which is always the smallest one.
At small frequencies, it is observed that the main influence of a chocked nozzle is the increase
of noise (ŵ + 2
↗) due to entropy waves, as already stated in [47]. This can be seen in the value
of T SA (ŵ + 2 /ŵS 1 ) which is as twice as large when the nozzle is chocked. The reflected wave ŵ− 1
increases also, but to a lesser extent. On the other hand, the acoustic response of the nozzle due
to an incoming acoustic wave remains quite similar (T AA and R AA ) for both unchocked and
chocked configurations.
When the reduced frequency increases, the value of the acoustic coefficients starts to move
away smoothly from the analytical results. Up to Ω ≈ 2 results given by both numerical
solvers can be considered similar. At higher frequencies, the 3D Euler solver curves representing
T AA and R AA start to decrease faster, the largest differences appearing for R AA and the
unchocked case. This is due to the numerical scheme used in the 3D Euler solver, which appears
to be dissipative for small length waves. Recall that the mesh is kept unchanged when
Ω increases, so that numerical errors increase. On the other hand, a higher spatial resolution is
considered in the quasi-1D LEE solver (2000 cells) and the corresponding results are virtually
free of numerical errors (it has been checked that grid convergence is reached for the results
displayed).
6.5 When entropy does not remain constant through a duct
In the previous section the influence of the inlet/outlet Mach number ¯M on the transmitted
and reflected waves was studied in isentropic nozzles. The aim of this section is not anymore to