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

3.3 Acoustic network model
p ′ (x, t )
= exp(i ω t ) ( f exp(−i k + x)+ g exp(+i k − x) ) , (3.31)
ρ 0 c
u ′ (x, t )=exp(i ω t ) ( f exp(−i k + x)− g exp(+i k − x) ) . (3.32)
The term exp(i ωt ) is omitted in the following as the harmonic time dependence
is the same for every location in space.
3.3 Acoustic network model
An acoustic low-order network model is a fast, flexible <strong>and</strong> useful tool to analyze
the dynamic properties <strong>of</strong> a complex system. In a network model the
entire system is divided into discrete acoustic elements. In this way a complex
system becomes easy to h<strong>and</strong>le. The acoustic characteristics <strong>of</strong> each element
are usually represented by its transfer matrix. A transfer matrix describes the
relationship <strong>of</strong> the acoustic pressure <strong>and</strong> the velocity or the Riemann Invariants
f <strong>and</strong> g between the inlet <strong>and</strong> the outlet <strong>of</strong> the element. Acoustic network
models are widely used in the context <strong>of</strong> thermo-acoustic analysis <strong>of</strong> gas turbines
<strong>and</strong> are described in many publications [10,20,23,46,58,61,70] (see also
chapter 2.1). The presented network model approach is based on the work by
Keller [53] <strong>and</strong> Polifke et al. [101, 103]. Applications <strong>of</strong> this approach can be
found in Huber <strong>and</strong> Polifke [49]. A sketch <strong>of</strong> a network model representing a
combustion system is shown in Fig. 3.2.
The mathematical bases <strong>of</strong> acoustic network models are the linear acoustic relations
derived in the last section, especially the wave equation, the solutions
<strong>of</strong> d’Alembert <strong>and</strong> the Bernoulli equation. The elements <strong>of</strong> the acoustic system
in the present work are mainly characterized by two in-ports <strong>and</strong> two outports,
whereas the boundary conditions possess one in- <strong>and</strong> one out-port. An
exception are "T-junctions" in form <strong>of</strong> a ”joint” or ”fork” with altogether four
in-ports <strong>and</strong> two out-ports or two in-ports <strong>and</strong> four out-ports, respectively. A
”joint” represents an intersection, in which two ducts are combined into one,
whereas a ”fork” is an intersection, in which a duct is divided into two. The in<strong>and</strong>
out-ports <strong>of</strong> an element are connected by the Riemann Invariants f <strong>and</strong>
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