Impact of fuel supply impedance and fuel staging on gas turbine ...

Acoustics
Beside the transfer matrix for variables f <strong>and</strong> g , the scattering matrix <strong>and</strong> the
transfer matrix characterized by p <strong>and</strong> u are widely used. Considering the punotation,
f <strong>and</strong> g are replaced by the normalized acoustic pressure p ′ /(ρ 0 c)
<strong>and</strong> the velocity u ′ . The scattering matrix, on the other h<strong>and</strong>, describes the
acoustic behavior <strong>of</strong> an element in a pure causal way. The relationship between
the Riemann Invariants f <strong>and</strong> g is determined by the direction <strong>of</strong> wave
propagation (see Fig. 3.4):
[ ] f j
≡ [ ] [ ]
f i
S sr . (3.34)
g i g j
Here, f i <strong>and</strong> g j denote the input or the signal whereas f j <strong>and</strong> g i represent the
output or the response. All different forms <strong>of</strong> matrices can be mathematically
transformed into each other as described in appendix A.1.
In some cases it is useful for a better underst<strong>and</strong>ing to split the transfer matrix
in matrices for each pair <strong>of</strong> ports. This is applied in the following to the ”Tjunction”,
the flame <strong>and</strong> area change element.
Transfer functions, on the other h<strong>and</strong>, connect one acoustic variable with another
as in the causal input-output relationship <strong>of</strong> the flame transfer function
or <strong>of</strong> the causal (reflection coefficient R f ) <strong>and</strong> non-causal (acoustic
<strong>impedance</strong> Z ) description <strong>of</strong> the boundary conditions. Mathematically the
<strong>impedance</strong> is defined as the acoustic pressure p ′ divided by the acoustic velocity
component normal to a surface u ′ n :
Z (ω)= p′ (ω)
(3.35)
u n ′ (ω).
The <strong>impedance</strong> is commonly used to characterize the propagation (transmission,
absorption) <strong>of</strong> sound through a medium, or the reflection <strong>of</strong> sound at
the boundary <strong>of</strong> two materials having different acoustic <strong>impedance</strong>s. In the
present work it is used to describe the acoustic characteristics <strong>of</strong> the <strong>fuel</strong> injection
stages. The reflection coefficient describes the relationship between
the reflected acoustic wave <strong>and</strong> the incident wave traveling towards a surface.
For the downstream side, for example, it yields:
R f (ω)= g (ω)
f (ω) . (3.36)
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