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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
6.5 When entropy does not remain constant through a duct 107 Unchoked Case ¯M2 =0.8 Choked Case ¯M2 =1.2 "! " ! "'()*+,-(.-,/*,0123+(45+6,- $'()*+,-(728,(45+6,- 903+:;21< "! " ! "'()*+,-(.-,/*,0123+(45+6,- $'()*+,-(728,(45+6,- 903+:;21< T AA T AA "! ! R AA "! ! R AA T SA "! !" T SA "! !" "! !# R SA R SA "! !# ! " # $ % & ! " # $ % & (a) (b) Figure 6.4: Reflection and Transmission coefficients for both unchoked and choked cases
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UNIVERSITE MONTPELLIER II SCIENCES
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An expert is a man who has made all
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Y gracias familia mía!!, que así
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Abstract Today, much of the current
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4 CONTENTS 3.4.1 Preconditioning .
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List of Figures 1.1 Main sources of
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8 LIST OF FIGURES 5.19 Acoustic ene
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List of Tables 1.1 EU recommended r
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14 Chapter 1: General Introduction
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2 Computation of noise generated by
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22 Chapter 2: Computation of noise
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