THESE de DOCTORAT - cerfacs

Chapter 7: Computation of the Reflection Coefficient on the Inlet air circuit of an Helicopter combustor
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chamber
respect to real(Y), imag(Y). The Neumann condition u ′ = 0 corresponds to a case in which
|R| = 1 and Arg(R) = 0, which in turn is equivalent to real(Y) = 0 and imag(R) = 0.
Changing the phase of R while maintaining |R| = 1 is equivalent to change imag(Y) while
keeping real(Y) = 0. On other hand, if Arg(R) is fixed to zero but |R| changes, it corresponds
to a variation of real(Y) maintaining imag(Y) = 0. By doing so, it is possible to compute the
correspondant resonant frequencies for the following hypothetical and extreme cases:
• The compressor does not change the phase of the acoustic waves but only their amplitude
ratio, i.e., Arg(R) = 0 and |R| varies from 0.1 to 1000.
• The compressor does not change the amplitude of the waves but only produce a phase
shift, i.e., |R| = 1 and Arg(R) varies from −π to π.
5
4
3
|R| = 0.2
|R| = 0.5
|R| = 1.0
|R| = 2.0
|R| = 5.0
5
4
3
|R| = 0.2
|R| = 0.5
|R| = 1.0
|R| = 2.0
|R| = 5.0
2
2
1
1
real Y
0
imag Y
0
−1
−1
−2
−2
−3
−3
−4
−4
−5
−3 −2 −1 0 1 2 3
Argument R
(a)
−5
−3 −2 −1 0 1 2 3
Argument R
(b)
Figure 7.7: Admittance vs. Reflection Coefficient
From Fig. 7.8 it is understood that a considerable variation is produced on the resonant frequency
when the phase is shift at the inlet of the combustor. This frequency can vary by 100Hz
when modifying the phase from −π to π. On the contrary, varying the modulus of the reflec-