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Thermoacoustic Instabilities in a Gas
Turbine Combustor
The Royal Institute of Technology
School of Technological Sciences
Department of Vehicle and Aeronautical Engineering
The Marcus Wallenberg Laboratory for Sound and Vibration Research
Karl Bengtson, karlbeng@kth.se
Acknowledgements
This report summarizes my study of thermoacoustic instabilities in a gas turbine combustor performed
at Siemens Industrial Turbomachinery AB during the autumn of 2017. The work was performed as
the final part of the Master’s program in engineering mechanics at the Royal Institute of Technology,
Stockholm, Sweden.
I would like to thank my supervisor Dr. Jan Pettersson who has encouraged and guided me through
the work at Siemens. I have learned a lot even though there are still much to learn and investigate
within the complex world of thermoacoustics. Many thanks to all other colleagues at Siemens for
support and especially to Joachim Nordin and Anders Haggmark who gave me the opportunity to
perform this work at Siemens.
I would also like to send my appreciation to Prof. Mats Abom at the Royal Institute of Technol-
ogy for being my supervisor at the university.
Last but not least I would like to thank my fiancee Madeleine for support, patience and under-
standing throughout my master studies.
Karl Bengtson
Finspang 2017-12-08
I
Abstract
Stationary gas turbines are widely used today for power generation and mechanical drive applications.
The introduction of new regulations on emissions in the last decades have led to extensive develop-
ment and new technologies used within modern gas turbines. The majority of the gas turbines sold
today have a so called DLE (Dry Low Emission) combustion system that mainly operates in the lean-
premixed combustion regime. The lean-premixed regime is characterized by low emission capabilities
but are more likely to exhibit stability issues compared to traditional non-premixed combustion sys-
tems.
Thermoacoustic instabilities are a highly unwanted phenomena characterized by an interaction be-
tween an acoustic field and a combustion process. This interaction may lead to self-sustained large
amplitude oscillations which can cause severe structural damage to the gas turbine if it couples with
a structural mode. However, since a coupled phenomena, prediction of thermoacoustic stability is a
complex topic still under research.
In this work, the mechanisms responsible for thermoacoustic instabilities are described and a 1-
dimensional stability modelling approach is applied to the Siemens SGT-750 combustion system.
The complete combustor is modelled by so called acoustic two-port elements in which a 1-dimensional
flame model is incorporated. The simulations is done using a generalized network code developed
by Siemens. The SGT-750 shows today excellent stability and combustion performance but a deeper
knowledge in the thermoacoustic behaviour is highly valued for future development.
In addition, measurement data from an engine test is evaluated, post-processed and compared with
the results from the 1-dimensional network model. The results are found to be in good agreement and
the thermoacoustic response of the SGT-750 is found to be dominated by both global modes including
all cans as well as local modes within the individual cans.
II
Contents
Acknowledgements I
Abstract II
Contents III
Nomenclature V
1 Introduction 1
1.1 Gas Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Siemens Gas Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 The SGT-750 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Thermoacoustic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.6 Thesis Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Theory and Fundamental Concepts 5
2.1 Combustion Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Classifications of Flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Equivalence Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.3 Adiabatic Flame Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Combustion Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Rayleigh’s Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 The Rijke Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.3 Driving Mechanisms for Thermoacoustic Instabilities . . . . . . . . . . . . . . . 10
2.2.4 Eigenmodes in Gas Turbine Combustors . . . . . . . . . . . . . . . . . . . . . . 11
2.2.5 Non-linear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Acoustic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 The Linearized Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 The Convective Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.3 Solutions to the Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.4 Impedance and Reflection Coefficient . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Damping of Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1 The Heat Release Source Term . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Acoustic Network Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5.1 The Straight Duct Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5.2 The Area Discontinuity Element . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5.3 Thermoacoustic Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Flame Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6.1 Acoustic Jump Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
III
2.6.2 The n− τ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6.3 An Extended Flame Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6.4 Fuel Injection Time Lag τi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6.5 Distributed Flame Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.7 Boundary Conditions at Combustor In- and Outlet . . . . . . . . . . . . . . . . . . . . 28
3 Method and Numerical Tool 31
3.1 Network Modelling Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Sample Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.1 Eigenfrequencies for a Simple Duct . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.2 Wave Transmission through an Expansion Chamber . . . . . . . . . . . . . . . 32
3.2.3 Rijke Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.4 Simple Burner Featuring a Sudden Expansion . . . . . . . . . . . . . . . . . . . 36
4 Application to the SGT-750 42
4.1 Evaluation of Measurement Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 Stability Analysis by the Network Modelling Approach . . . . . . . . . . . . . . . . . . 45
4.2.1 Establishing the Acoustic Network Model . . . . . . . . . . . . . . . . . . . . . 45
4.2.2 The Influence of a Mean Flow on the Acoustics . . . . . . . . . . . . . . . . . . 48
4.2.3 Introducing the Flame - Perfectly Premixed Case . . . . . . . . . . . . . . . . . 49
4.2.4 The Outlet Acoustic Boundary Condition . . . . . . . . . . . . . . . . . . . . . 52
4.2.5 Fuel Line Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.6 Equivalence Ratio Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.7 Utilizing the Full Flame Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Measures to Improve Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3.1 Change Combustor length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3.2 Including Helmholtz Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.3 C-stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4 Conclusions from the SGT-750 Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5 Discussion 66
6 Recommended Future Work 67
APPENDIX
A The Two Microphone Method
IV
Nomenclature
Lower case letters
c - Speed of sound
f - Frequency
hf - Specific reaction enthalpy
i - Imaginary unit (i =√−1)
k - Wave number (k = ω/c)
k+ - Wave number for waves propagating in the positive direction
k− - Wave number for waves propagating in the negative direction
m - Mass
m - Mass flow rate
p - Pressure
q - Heat release density
s - Entropy
t - Time
u - Velocity magnitude
u - Velocity vector
x - Spatial coordinate (1-D)
x - Spatial coordinate vector (3-D)
yf - Mass fraction of fuel
z - Characteristic impedance (z = ρc)
Upper case letters
A - Area
C+ - Amplitude of wave propagating in positive direction
C− - Amplitude of wave propagating in negative direction
Cxy - Coherence between signal x and y
D - Diameter
F - Flame transfer function
Hxy - Complex transfer function (Hxy = x(ω)/y(ω))
L - Length
M - Mach number (M = u/c)
Pxx - Auto power spectrum of signal x
Pxy - Cross spectrum between signal x and y
Q - Heat release
R - Wave amplitude reflection coefficient
T - Period of oscillation (T = 1/f)
Tij - Transfer element ij
T - Transfer matrix
V
V - Volume
V - Volume flow rate
Z - Specific impedance
Greek letters
γ - Ratio of specific heats (γ = Cp/Cv)
φ - Equivalence ratio
ρ - Density
ω - Angular frequency (ω = 2πf)
τ - Time lag
τb - Time lag related to volume flow fluctuations
τi - Time lag related to equivalence ratio fluctuations
∇ - Nabla operator, ∇ = (∂/∂x, ∂/∂y, ∂/∂z)
Notations
p∗ - Complex conjugate
p - Mean value of quantity
p′ - Fluctuating part of quantity (Acoustic part)
p - Complex amplitude - Frequency domain representation
VI
1 Introduction
Gas turbines are used all around the world for power generation and as power sources for pumps and
compressors. The rapid development and construction of renewable energy sources such as wind and
solar energy has created a new demand for flexible power sources that in combination constitutes a
reliable energy system. Gas turbines have the ability to start up quickly when the sun is not shining
or the wind is not blowing. For this reason, gas turbines will be a central part within the power
generation business for many years to come.
The development of the next generation gas turbines primarily focuses on improved efficiency and
reduced NOx (nitrous oxides) emissions. The majority of the stationary gas turbines produced today
have a DLE (Dry Low Emission) combustion system to meet emission regulations. DLE combustion
systems operate primarily in the premixed combustion regime and are more likely to experience com-
bustion instabilities than conventional combustors.
There are several measures used today to reduce NOx emissions in gas turbines. The most efficient
option is to lower the combustion temperature. However, this is in general counter-productive when
it comes to engine efficiency and the general trend within thermal machinery is instead to increase
temperatures in order to improve engine efficiency. Another way to reduce emissions often utilized is
different measures to improve combustion stability allowing for reduction of the stabilizing diffusion
pilot and hence lower NOx emissions. To improve combustion stability, understanding and prediction
of thermoacoustic instabilities is an important key.
Thermoacoustic instabilities are characterized by an interaction process between a combustion pro-
cess and an acoustic field which may lead to self-sustained oscillations. This oscillations may grow
in amplitude and can cause wear and severe structural damage if not kept below acceptable limits.
Since thermoacoustic instabilities is formed from the coupling between combustion and acoustics, ac-
curate modelling need to include both phenomena which makes thermoacoustic modelling complex
and difficult to deal with. However, for a successful combustion system, thermoacoustic instabilities
need to be understood and predicted. This will be even more important in the future with tougher
regulations on emissions as well as new demands for flexible operation.
A frequently used modelling approach for thermoacoustics is by using so called 1-dimensional low-
order models. This work is one attempt to model and predict thermoacoustic instabilities for a gas
turbine combustor using a low-order network approach.
1.1 Gas Turbines
Gas turbine engines work according to the Brayton cycle and can be divided in three main parts,
compressor, combustor and turbine. Compressed air from the compressor enters the combustor where
fuel is introduced and the mixture is burnt. The combustion gases are expanded through the turbine
1
which drives the compressor as well as giving a net power output that can be used to drive a generator
or a pump. The main fuel used in modern gas turbines are natural gas but many engines have the
capability to operate on liquid fuel as well as alternative gas mixtures. Only a brief description of
the gas turbine working principle is given here, more details can be found in e.g. the gas turbine
handbook by Boyce [4].
Two different types of gas turbine combustors are commonly used in modern gas turbines. These
are can and annular type combustors. In a can combustor system, each burner has its own combus-
tion chamber while in an annular system all burners goes into one large annular combustion chamber.
Annular combustors in general have less area to cool while the can combustors have advantages when
it comes to service and maintenance. Also, in annular combustors the burner-to-burner interaction is
in general more apparent than for can combustors.
1.2 Siemens Gas Turbines
Siemens is a well known manufacturer of industrial gas turbines with a portfolio including gas turbines
in the range from 4 to 400MW,[26]. The Siemens gas turbines are sold to customers all around the
world while development and manufacturing are at present concentrated mainly to Europe and North
America. Siemens Industrial Turbomachinery AB in Finspang, Sweden is responsible for the industrial
mid-size engines. The turbine production in Finspang started in 1913 and since then more than 800 gas
turbines have been manufactured and delivered. The Finspang site has today about 2700 employees
and the site is responsible for development of both core and package, production, sales, commissioning
and after-market/service of their engines, [26].
1.3 The SGT-750
The SGT-750 is one of the latest industrial gas turbines from Siemens with a power output of around
40MW and world class efficiency. The engine has a twin-shaft configuration meaning the compressor
and power turbine are disconnected from each other, this is especially suitable for mechanical drive
applications. The SGT-750 core engine is shown in figure 1.
2
Figure 1: The SGT-750 core engine, [26].
The SGT-750 combustor features eight individual cans which provides single digit NOx capabilities in
a wide range of operation conditions. One of the eight combustor cans is shown in figure 2. Each of the
combustor cans are fed by compressor discharge air through a common annular casing. Convective and
impingement cooling techniques are utilized for cooling of the combustion chamber walls. The burner
comprises two separate main fuel lines (main 1 and main 2) for further improved tuning flexibilities.
An optimized aerodynamic design ensures a well-defined recirculation zone for stabilizing the flame.
In addition, a pilot and a RPL (Rich Pilot Lean) burner are used for central stabilization of the main
flame. After combustion, the combustion gases are led through a transition duct to the inlet of the
first turbine stage.
Figure 2: The SGT-750 combustion system, [13].
3
1.4 Thermoacoustic Modeling
Many studies with different modelling approaches to thermoacoustics in gas turbines and jet-engines
can be found in the literature. Those includes forced response analysis, complex eigenvalue analysis,
URANS (Unsteady Reynolds Averaged Navier Stokes) and LES (Large Eddy Simulations). Even
though thermoacoustics is a coupled phenomenon, many studies focus more on either combustion or
acoustics. However, the necessity of including the coupling and e.g. complementing LES with acoustic
analysis methods is pointed out by Poinsot, [22]. A common used approach to study stability in a
coupled manner is to use a so called low-order network model. Such model assumes 1-dimensional
acoustics and uses elements known as two-ports to describe the combustor geometry and the flame.
Together with up- and downstream boundary conditions a stability criterion can be formulated and
analysed. This is the approach used in this work.
1.5 Thesis Objectives
The objectives with this work were to:
• Perform a literature study on thermoacoustic stability analysis and low-order modelling.
• Evaluate and understand a general network code developed by Siemens AG.
• Create a thermoacoustic network model of the SGT-750 combustion system.
• Evaluate and post-process combustion dynamic measurement data from full engine tests.
• Compare the results and give recommendation of future work.
1.6 Thesis Motivation
The main objective with thermoacoustic stability analysis is to predict the systems dynamic behaviour
early in the design phase. Anyhow, due to the complex coupled phenomena, this has been found to be
easier said than done. Using different modelling techniques is one way to gain understanding in the
field and to make justified design changes to improve stability. In this thesis work, one approach to
model thermoacoustic stability is utilized. The SGT-750 shows today excellent combustion stability
and performance. However, a better understanding of the dynamic behaviour is beneficial for further
development and possible future upgrades of the engine.
4
2 Theory and Fundamental Concepts
In this section, concepts and relevant theory needed to understand combustion instabilities are pre-
sented. Since the combustion process itself is the driver for the instabilities a review of some funda-
mental combustion concepts is given first.
2.1 Combustion Fundamentals
Combustion is an exothermal reaction process characterized by conversion of chemical spices and heat
release. A combustion process involves fuel and an oxidizer which for gas turbines commonly are
hydrocarbons and air. The chemical reaction when methane is burnt in air on a global level, not
considering intermediate species can be expressed as
CH4 + 2O2 ⇒ CO2 + 2H2O +Heat. (1)
The heat release within a combustion process appears as a flame which propagates through the
unburned mixture with a certain burning velocity. The flame can be defined as a thin layer with
rapid chemical changes and a steep temperature gradient. On a macroscopic level, the flame is often
seen as the interface that divides the burned mixture from the unburned mixture. In the science
of combustion, both physics and chemistry play important roles, [17]. To describe heat release in
thermoacoustic studies the physics and thermodynamics is often sufficient while a detailed chemistry
description is more vital when e.g. emissions are to be predicted.
2.1.1 Classifications of Flames
Flames in general can be classified as either premixed or non-premixed flames. In addition, flames
can also be categorized as laminar or turbulent depending on the initial state of the reactants, [28].
Both premixed and non-premixed turbulent flames are utilized in modern gas turbines.
For premixed flames the fuel and oxidizer are mixed before entering the flame zone while for non-
premixed flames the fuel and oxidizer are mixed by diffusion within the flame zone. Premixed flames
are in general more sensitive to disturbances and stability issues than non-premixed flames. The main
reason is that non-premixed flames stabilizes in the intermediate mixing region between the fuel and
the oxidizer which constitutes steep concentration gradients, [28]. On the other hand, for a perfectly
premixed mixture, no such concentration gradients exist and hence no obvious location for the flame
to stabilize. This makes the flame location very sensitive to disturbances. Combustors operating in
the premixed regime generally features a sudden expansion where the combustible mixture enters the
combustion chamber. The velocity before the expansion is high enough to prevent the flame from
propagating upstream. This will create a defined location for the flame to stabilize. A schematic
picture of general combustor featuring a sudden expansion is shown in figure 3.
5
Figure 3: Schematic picture of a simple burner featuring a sudden expansion.
Gas turbine combustors featuring DLE combustion systems utilize in general both premixed and non-
premixed flames. The largest portion of the fuel is introduced in the main flame which is a lean
premixed flame that produces low levels of NOx emissions. Ideally the main combustible mixture is
homogeneous but this is difficult to realize in practice and mixing performance is a topic of continuously
development and research. In addition to main fuel, some smaller portion of the fuel is introduced as
a pilot which gives a fuel-rich often close to non-premixed flame. The pilot flame is used to stabilize
and maintain the main flame but has the drawback to produce higher levels of NOx emissions. With
improved combustion stability the PFR (Pilot Flow Rate) can be reduced and hence NOx emission
levels are reduced.
2.1.2 Equivalence Ratio
Equivalence ratio describes the fuel-to-air ratio and is an important parameter within the science of
combustion. Equivalence ratio is defined as
φ =mfuel/mox
(mfuel/mox)st. (2)
The index st denotes stoichiometric conditions which is the situation when the amount of air is
exactly what is needed to completely burn the fuel. Hence, stoichiometric conditions correspond to
an equivalence ratio equal to 1. An equivalence ratio higher than 1 means there is an excess of fuel to
the available amount of oxidizer, this is called fuel-rich and will give unburnt fuel as a rest product.
If the equivalence ratio is less than 1 there is an excess of air to the amount of fuel to be burnt. This
condition is called fuel-lean, [28].
2.1.3 Adiabatic Flame Temperature
Flame temperature has a strong influence on the chemical reaction taking place within a flame and
are strongly coupled to both emissions and heat release. The maximum flame temperature is achieved
at stoichiometric conditions. Adiabatic flame temperature is a quantity that can be calculated from
the thermodynamic properties of the reactants. It is defined as the temperature that the net energy
6
released in the flame would give to the combustion products under adiabatic conditions. A real com-
bustion process is not adiabatic and the actual flame temperature is always lower than the adiabatic
flame temperature. The adiabatic flame temperature is mainly affected by equivalence ratio, initial
temperature and pressure, [28].
2.2 Combustion Instabilities
Combustion instabilities can be divided in two categories, Combustion noise and Thermoacoustic
instabilities. Those are both driven by the combustion process but the characteristics and physical
phenomena is different.
Combustion Noise
The flow in gas turbine combustors is inherently turbulent. This turbulence creates flow variations
that affects the combustion process and results in combustion noise. This noise is sometimes called
”combustion roar” and is of a broadband character with relatively low amplitude, [9]. Combustion
noise is not that critical in modern gas turbines and has not been treated in this work.
Thermoacoustic Instabilities
Thermoacoustic instabilities on the other hand commonly appears as large amplitude oscillations at
one of the systems natural frequencies. Those instabilities are spontaneously excited and the oscilla-
tions are maintained by a feedback loop between the combustion and the acoustic field. The principle
for thermoacoustic instabilities is illustrated in figure 4 and 5. The unsteady heat release in the flame
generates acoustic waves which are reflected at the system boundaries and standing waves are formed.
The acoustic fluctuations give rise to flow and mixture perturbations which in turn affects the flame
with a fluctuation of the heat release as the result, the loop is closed, [19].
Figure 4: Acoustic waves are created by the flame and reflected at the system boundaries.
The oscillations will be amplified or damped depending on the phase between the heat release and
the pressure. In contrast to combustion noise, thermoacoustic instabilities are characterized by high
amplitude oscillations at distinct frequencies. Those large oscillations in velocity and pressure are
highly unwanted and can cause severe wear and structural damage to the gas turbine.
7
Figure 5: The feedback loop responsible for thermoacoustic instabilities, [19].
Combustion processes can create acoustic waves in at least two different ways. First, acoustic waves
will be created directly by the volume fluctuations resulting from the unsteady heat release. Ad-
ditionally, unsteady heat release gives rise to temperature fluctuations often refereed to as entropy
waves which convect downstream with the mean flow. Those entropy waves are not associated with
any acoustic fluctuation of pressure and velocity, hence no noise. However, when an entropy wave is
accelerated as happens at the combustor exit, acoustic waves will be indirectly generated. Entropy
waves and indirect noise has become an important topic for aero-engines to reduce the overall noise
level, [9]. The effect of entropy waves and indirect noise on thermoacoustic instabilities is an ongoing
discussion and have been discussed by e.g. Goh and Morgans, [10] and Sattelmayer, [6].
2.2.1 Rayleigh’s Criterion
The situation under which thermoacoustic instabilities occurs was first described by Lord Rayleigh in
the 1880s. His criteria is important for the understanding of thermoacoustic instabilities and can be
formulated as, [19],
ˆV
ˆT
p′(x, t)q′(x, t)dtdV > 0. (3)
Energy is transferred to the acoustic field if the phase difference between the unsteady pressure and
heat release is less than 90◦ leading to amplified oscillations. Maximum energy input to the acoustic
field is archived if pressure and heat release are perfectly in phase. If on the other hand the heat
release and the pressure is out-of-phase, energy is removed from the acoustic field and the oscillations
are damped. Rayleigh’s criteria as formulated in equation 3 is strictly valid for undamped systems.
For real systems, some portion of the acoustic energy propagates out through the boundaries or gets
dissipated by friction and viscous effects. For a system that includes damping, the Rayleigh integral
need to render a larger value than the energy dissipated to get amplified oscillations, [19].
8
2.2.2 The Rijke Tube
A classical experiment that has highly contributed to the understanding of thermoacoustic instabilities
is the Rijke tube. In its most basic form, a Rijke tube constitutes an open-ended pipe with a heat
source which under certain conditions will generate a strong tonal sound. For sound to be generated
a mean flow is required which can be created by convection if placing the tube vertically. The heat
source is commonly an electrically heated grid or a combustion flame, e.g. a Bunsen burner. Figure
6 shows a schematic representation of a Rijke tube.
Figure 6: Schematic representation of a Rijke tube
The phenomena generating the sound can be understood from the Rayleigh’s criteria (equation 3).
The first mode of an open-ended tube has pressure nodes at the ends and a velocity node in the middle
as shown in figure 6. As can be seen the pressure has the same sign (positive) in the whole tube while
the velocity changes sign in the middle of the tube. Considering an electrically heated grid, the heat
release from the source to the medium in the tube will be due to convection. The convection process
is influenced by the velocity at the grid which will be the mean velocity superimposed by the acoustic
fluctuations. The unsteady heat release by the source is obviously related to the acoustic velocity
fluctuations and the same is true for a combustion heat source. Placing the heat source in the lower
half of the tube will render a positive value of the Rayleigh’s integral and the acoustic oscillations will
be sustained and amplified. In contrary, placing the heat source in the upper part of the tube the
9
Rayleigh’s integral will give a negative value and the acoustic oscillations will instead be attenuated,
[2].
For an acoustic field without a source, pressure and velocity fluctuations are 90 degrees out-of-phase.
This means, if the heat release is perfectly in-phase with the velocity, no energy would be transferred
to the acoustic field according to Rayleigh’s criteria. This gives a second criteria for instability. It
must be a delay between the heat release and the acoustic velocity oscillations. Referring to the
electrical heater the heat release is due to convection which is a process that takes time. The same is
true for a combustion process were some time is required for the chemical reaction to take place.
The tone generated by a Rijke tube is normally the fundamental tone, with a wavelength corre-
sponding to twice the length of the tube. However, the frequency is not easy to predict in practice.
First, the phase of the heat release at the heat source location will influence the acoustic impedance
which will influence the eigenfrequencis, this has e.g. been studied by Mcintosh, [20]. In addition, due
to the heat source, there will be a complex temperature distribution in the tube affecting the speed
of sound and hence the eigenfrequencies. The influence of the temperature field in a Rijke tube has
been studied by L.Nord, [21].
2.2.3 Driving Mechanisms for Thermoacoustic Instabilities
There are several mechanisms responsible for driving of thermoacoustic instabilities in gas turbines.
This section gives a description of some of the most important mechanisms.
Equivalence Ratio Fluctuations
Equivalence ratio fluctuations are an important source to combustion instabilities in premixed com-
bustors operating at fuel-lean conditions. Acoustic perturbations within the premixing section may
influence the air and/or fuel supply leading to periodic equivalence ratio oscillations. Those fluc-
tuations are convected by the mean flow to the flame front resulting in an unsteady heat release.
Equivalence ratio fluctuations due to acoustic coupling is strongly affected by the pressure drop over
fuel injectors. In general, a larger pressure drop makes the system less sensitive to acoustic distur-
bances, [9].
Coupling Acoustic-Fuel Feed Line
Fuel feed line-acoustic coupling is a special case of equivalence ratio fluctuations. The mechanism
refers to pressure drop fluctuations over non-chocked nozzles which makes the fuel injection rate to
be modulated. The origin of the pressure drop fluctuations can be both on the fuel line or due to
the acoustic field in the combustor, [9]. The later case is illustrated in figure 7. The flame generates
acoustic waves leading to pressure fluctuations over the fuel nozzles located upstream. This imply
the amount of fuel injected will vary periodically in time and the resulting mixture is convected to
the flame front. The time required for the disturbance to convect to the flame front is here denoted,
10
τi. Depending on the phase between the unsteady heat release and the pressure at the flame front,
the oscillation will be amplified or damped. It can be concluded that convective times are important
parameters for control of thermocoustic instabilities.
Figure 7: Coupling acoustic-fuel feed line.
Flame Area Variation
The heat release at the flame front is proportional to the flame surface area. There are several reasons
the flame area may vary in gas turbine combustors leading to a fluctuating heat release. The most
important mechanisms are:
• Acoustic velocity oscillations within the combustor will affect the flame area.
• Flame-vortex interaction which refers to periodic separations created by e.g. sudden expansions,
flame holders or other obstacles in the flow path. The vorticity generated convects with the flow
and stretches the flame when passing through leading to periodic variations of flame area. The
frequency at which this occur does not need to be the natural shedding frequency. If the
amplitude is high enough, vortex separation can be forced to occur by an external excitation.
• Vortex interaction with boundaries. Vortexes generated interact with a wall which periodically
introduce fresh unburned mixture into the flame zone. The fresh mixture ignites after some time
delay and creating a fluctuating heat release.
• Thermal losses when a flame impinges on a cold wall can affect the chemical reaction and make
the flame area to vary.
• Interaction between flames may affect the flame surface area. This could be between pilot and
main flame as well as between different burners in annular combustion systems.
2.2.4 Eigenmodes in Gas Turbine Combustors
Thermoacoustic instabilities is normally associated with one or more acoustic eigenmodes of the
system. Gas turbine combustors generally features cylindrical geometries with hard metal walls.
Possible mode shapes are illustrated in figure 8.
11
Figure 8: Modes in combustor geometries. a) Longitudinal mode. b) Transverse Azimuthal Mode. c)
Transverse Radial Mode.
The longitudinal modes are to a large extent controlled by the boundary conditions at the combustor
inlet and outlet while the transverse modes are more defined due to the hard metal walls. It is assumed
the interaction with the walls is negligible. No transverse modes can exist bellow the cut-on frequency
for the first higher order mode. The cut-on frequency for the first transverse azimuthal mode in a
circular geometry is given by
f1,cut−on = 1.841c
πD. (4)
In the low frequency range, below the cut-on frequency for the first higher order mode, the acoustic field
is more or less one-dimensional. Typical temperatures in gas turbine combustors are 750K upstream
the flame (compressor discharge air temperature) and 1750K downstream of the flame which gives
the speed of sound to be about 550m/s and 830m/s respectively.
2.2.5 Non-linear Effects
According to Rayleigh’s integral, energy will be added to the acoustic field as long as the integral is
larger than the dissipation. The amplitude of the oscillations will grow exponentially in the beginning
but this cannot continue forever. Non-linear effects will cause the amplitude of the oscillations to sat-
urate at some finite limit-cycle amplitude. To determine the limit-cycle amplitude, non-linear effects
cannot be neglected and hence, the limit-cycle amplitude cannot be determined by linear models, [19].
Linear models however are able to predict potential critical frequencies. In general the acoustic quan-
tities are small enough to be treated as linear while the heat release includes non-linear phenomena.
Taking advantage of this have led to hybrid models where linear acoustics is coupled to non-linear
heat release models, this approach has been used by e.g. Graham and Dowling, [11].
Only linear models will be used through out this work.
12
2.3 Acoustic Theory
2.3.1 The Linearized Wave Equation
The most fundamental equation within acoustics is the linearized wave equation. The derivation starts
from the well known conservation equations from fluid mechanics. These are conservation of mass,
momentum and energy which in differential form may be written as, [25],
∂ρ
∂t+∇ · (ρu) = m, (5)
ρ
[∂u
∂t+ u · ∇u
]+∇p = f , (6)
ρT
[∂s
∂t+ u · ∇s
]= q. (7)
Where m,f , q are source terms representing mass sources, external forces and heat sources respec-
tively. Viscous effects are neglected.
Considering acoustic wave propagation in a homogeneous media without any sources. Zero mean
flow is assumed at this point. In most cases the acoustic perturbations can be assumed to be small
in comparison to the mean value. The acoustic field variables are described by a steady mean value
plus a small fluctuating part as
p(x, t) =p+ p′(x, t)
ρ(x, t) =ρ+ ρ′(x, t)
u(x, t) =0 + u′(x, t).
(8)
The overline denotes the mean value and the prime denotes the fluctuating (acoustic) part. Substitu-
tion of equation 8 into the conservation equations for mass and momentum and neglecting products
of primed quantities gives the linearized acoustic conservation equations as
∂ρ′
∂t+ ρ∇u′ = 0, (9)
ρ∂u′
∂t+∇p′ = 0. (10)
Since the acoustic disturbance being small a frequently used assumption is a sound wave being isen-
tropic and reversible. The relation between acoustic pressure and density can be found from the
equation of state, p = p(ρ, s) which may be expressed as
13
∂p
∂t=
(∂p
∂ρ
)s
∂ρ
∂t+
(∂p
∂s
)ρ
∂s
∂t. (11)
For an isentropic process the entropy is constant which makes the second term in equation 11 to
vanish. By using the definition of the speed of sound, c2 = (∂p∂ρ )s, the relation between the acoustic
pressure and density after linearization is found to be
p′ = ρ′c2. (12)
This relation is used to eliminate the density in the conservation equation for acoustic continuity, the
result is
∂p′
∂t+ ρc2∇u′ = 0. (13)
Now, taking the time derivative of Equation 13, the spatial derivative of equation 10 and subtracting
the two yields the linearized wave equation in 3-dimensions
∂2p′
∂t2− c2∇2p′ = 0. (14)
Through out this work, wave propagation is assumed to be 1-dimensional only and the wave equation
in 1-D is given below for reference.
∂2p′
∂t2− c2 ∂
2p′
∂x2= 0 (15)
It should be mentioned that the assumptions made in the derivation of the wave equation may be
appropriate for sound propagation in ambient temperature and pressure conditions. However, the
assumptions may not always be suitable for the conditions in a gas turbine combustor.
2.3.2 The Convective Wave Equation
The convective wave equation describes wave propagation when a mean flow is present. The effects
of a mean flow to acoustic wave propagation can be included in the previously derived wave equation
by a change of reference system. Derivation is straight forward and the result is found be replacing
the time derivative in equation 15 by the convective derivative, the result is
(∂
∂t+ u
∂
∂x
)2
p′ − c2 ∂2p′
∂x2= 0. (16)
14
2.3.3 Solutions to the Wave Equation
The solution to the wave equation in 1-dimension is a linear combination of two waves, one propagating
in the positive direction and one in the negative direction. If a mean flow is present, this positive
travelling wave propagates with a speed c + u while the negative travelling wave propagates with a
speed c− u. The well known d’Alembert’s solution can be written as
p′(x, t) = f(x− (c+ u)t) + g(x+ (c− u)t), (17)
with f and g being two arbitrary functions. A convenient description often used in acoustics is
obtained by assuming a harmonic time dependence for the acoustic quantities. The relation between
acoustic pressure and velocity for a plane wave is given by the characteristic impedance z = ±ρc,for the positive (+) and the negative (-) travelling wave respectively. Using this, the solution with
separated time and space dependence can be written as
p′(x, t) = C+e
i(ωt−k+x) + C−ei(ωt+k−x)
u′(x, t) =C+
ρcei(ωt−k+x) − C−
ρcei(ωt+k−x).
(18)
The wave number in the positive respective negative direction is given by
k+ =ω
c(1 +M), k− =
ω
c(1−M). (19)
If the mean flow being zero the wave number in both directions are reduced to k = ω/c. Acoustic
analysis is commonly performed in the frequency domain, the frequency domain solution to the wave
equation is obtained by a Fourier transformation which yields
p(x) = p+e
−ik+x + p−eik−x
u(x) =1
ρc
(p+e
−ik+x − p−eik−x).
(20)
Where p+ and p− are the complex amplitude of the waves propagating in the positive and negative
direction respectively.
2.3.4 Impedance and Reflection Coefficient
Since the acoustic theory described here is linear the ratio between the acoustic pressure and acoustic
velocity at any point will be independent of the sources. They will be related by the specific impedance
which in 1-D is defined as
Z =p(x, ω)
u(x, ω). (21)
15
For a plane wave in the direction of propagation, the specific impedance equals the characteristic
impedance (z = ρc). Impedance can also be used to describe transmissions and reflections at a
given section. The sign of the impedance depends if one are looking in the upstream or downstream
direction and careful use is required. A more intuitive way to describe wave reflections is by a reflection
coefficient defined as the ratio of the reflected wave amplitude to the incident wave amplitude. For an
upstream respective downstream boundary in 1-D, the reflection coefficient relates to impedance as
Rupstream =Z + ρc
Z − ρc, Rdownstream =
Z − ρcZ + ρc
. (22)
Where Z is a prescribed impedance at the boundary. Three important cases can be identified from
equation 22. For an acoustically hard wall (u′ = 0) which results in R=1 and for a soft wall such as an
open pipe (p′ = 0) and hence R=-1. For a non-reflecting boundary, impedance matching is required
(Z = ρc) and the refection coefficient is R=0.
2.4 Damping of Acoustic Waves
Even though gas turbine combustors in general are lightly damped, acoustic damping due to viscous
effects in the flow will be important at higher frequencies. For low frequencies the viscous effects are
small and may be neglected. Acoustic energy dissipation in gas turbine combustors due to viscous
effects can be divided in three categories, [7]. 1, Acoustic energy dissipation in boundary layers which
becomes important for narrow tubes and pipes. 2, Flow induced damping which refers to the dis-
sipation of acoustics energy in regions with strong vorticity generation such as area discontinuities.
Dissipation occurs due to acoustic energy is converted to turbulence within the vortexes. 3, Acoustic
energy dissipation in the free field, this damping is in general small compared to the other two damp-
ing mechanisms.
Damping of a propagating acoustic waves is normally included in models by a complex wave number
which accounts for the damping. Different corrections for damping can be found in the literature, one
way of including the damping in boundary layers can be found in [1].
Low frequencies are the topic of this work and damping effects due to viscous effects will be neglected.
2.4.1 The Heat Release Source Term
When a fluctuating heat source is present the isentropic assumption used in the derivation of the wave
equation without sources is not longer valid. This imply the second term in equation 11 do not longer
vanish and a new relation between acoustic pressure and density need to be established. The second
term in equation 11 can be rewritten by using the general gas law and the definition of entropy. The
equation of state can be expressed as, [25],
16
∂p
∂t= c2
∂ρ
∂t+ (γ − 1)
∂q
∂t. (23)
The heat release is assumed to consist of a steady and fluctuating part as q(x, t) = q + q′(x, t).
Linearization of equation 23 gives the relation between acoustic pressure and density when an unsteady
heat release source is present. The result is
∂p′
∂t= c2
∂ρ′
∂t+ (γ − 1)
∂q′
∂t. (24)
This result is used to eliminate the acoustic density in equation 9 and 10. The acoustic conservation
equations in 1-D become
∂p′
∂t+ ρc2
∂u′
∂x= (γ − 1)
∂q′
∂t, (25)
ρ∂u′
∂t+∂p′
∂x= 0. (26)
The wave equation is then derived in the same way as before, by taking the time derivative of equation
25, the spatial derivative of equation 26 and subtracting the two. The result is
∂2p′
∂t2− c2 ∂
2p′
∂x2= (γ − 1)
∂q′
∂t. (27)
Comparing the result with the wave equation without any sources (equation 15), the source term
representing unsteady heat release is found to be (γ − 1)∂q′
∂t .
2.5 Acoustic Network Modelling
For duct like systems such as gas turbine combustors, only plane waves can exist at sufficiently low
frequencies when the wave length is much longer than the geometrical dimensions of the cross-sections.
If only plane waves exist and the effect of coupled wall vibrations is negligible the system can be de-
scribed by so called acoustic two-port theory. Acoustic two-port theory is based on 1-dimensional
acoustics for which the acoustic field is fully determined by two field variables. Different formalisms
is used for two-ports in the literature but the principle is the same. In this work the acoustic field is
described by the acoustic pressure p′ and the acoustic velocity u′.
A two-port element relates the acoustic state at the inlet to the outlet by either analytical expres-
sions or measured transfer functions. In figure 9, a two-port element with inlet (a) and outlet (b) is
illustrated. The relation between the acoustic states, up- and downstream of the element is described
by a 2x2 transfer matrix (T) as given in equation 28. The strength in the two-port theory is that
complex systems can be modelled as a network or cascade of two-port elements.
17
Figure 9: Two-port element. The inlet (a) is related to the outlet (b) by the transfer matrix T.
(pb
ub
)=
(T11 T12
T21 T22
)(pa
ua
)(28)
The relation of the inlet to the outlet of a cascade of elements on the form as given in equation 28
can be obtained as a total transfer matrix as
Ttot =
N∏n=1
Tn, (29)
where N is the total number of elements. The two-port theory can be extended into a multi-port
formulation which can be used for modeling two-dimensional waves. E.g. 2-D multi-port elements for
wave propagation in annular cavities are described by [25].
The two most basic and used two-port elements in the modelling of gas turbine combustors are the
”straight duct” and ”area discontinuity” elements, the derivation of the analytical transfer functions
is given below.
2.5.1 The Straight Duct Element
The transfer matrix for a straight duct element is derived from the 1-D solution to the convective wave
equation. A duct oriented along the x-direction with length L as shown in figure 10 is considered.
Figure 10: The straight duct element.
18
The acoustic field inside the duct constitutes a left and a right travelling wave which can be described
as
p(x) = p+e
−ik+x + p−eik−x
u(x) =1
ρc
(p+e
−ik+x − p−eik−x).
(30)
The acoustic state at the inlet respective outlet are found by evaluating equation 30 at x = 0 and
x = L. This gives
pa = p(x = 0) = p+ + p−
ua = u(x = 0) =1
ρc(p+ − p−) ,
(31)
pb = p(x = L) = p+e
−ik+L + p−eik−L
ub = u(x = L) =1
ρc
(p+e
−ik+L − p−eik−L).
(32)
Putting these equations together and eliminating p+ and p− gives two equations relating the inlet
state to the outlet state. Formulation in the form of a transfer matrix yields
T =1
2
(e−ik+L + eik−L ρc
(e−ik+L − eik−L
)1ρc
(e−ik+L − eik−L
)e−ik+L + eik−L.
)(33)
2.5.2 The Area Discontinuity Element
The most simple way to model an area discontinuity is by using continuity relations. This means the
pressure must be the same on both sides of the discontinuity. Correspondingly the oscillating mass flow
must be conserved which gives the relation for acoustic velocity. For the linearized case the unsteady
part of the mass flow between the up- and downstream sections can be written as ρ1A1u′1 = ρ2A2u
′2.
Formulating this in the form of a transfer matrix yields
T =
(1 0
0 A1ρ1A2ρ2
.
)(34)
2.5.3 Thermoacoustic Stability Analysis
To analyse thermoacoustic stability of a system, the transfer matrix modelling approach is frequently
used. As before, the acoustic quantities are assumed to have a harmonic time dependence, that is
p′(t) = p(x)eiωt, u′(t) = u(x)eiωt. (35)
19
For stability analysis the angular frequency ω is allowed to be complex on the form
ω = Re(ω) + iIm(ω). (36)
From the assumed time dependence eiωt it is seen that the real part of the angular frequency represents
the oscillation while the imaginary part determines the rate of growth in time. If the imaginary part
is positive the amplitude of the acoustic quantities will decay in time and the oscillation is damped.
In contrary, if the imaginary part is negative the amplitude will grow in time and the system may
become unstable.
Consider a system consisting of 3 elements and with known impedance at the inlet and outlet as
outlined in figure 11. The system has 4 nodes with an acoustic state described by pi and ui at each
node, hence 8 degrees of freedom. By using the definition of impedance and the transfer matrix of
respective element, 8 equations can be formulated and assembled into a global system matrix as shown
in equation 37.
Figure 11: A simple network model for stability analysis.
−1 Zin 0 0 0 0 0 0
T(1)11 T
(1)12 −1 0 0 0 0 0
T(1)21 T
(1)22 0 −1 0 0 0 0
0 0 T(2)11 T
(2)12 −1 0 0 0
0 0 T(2)21 T
(2)22 0 −1 0 0
0 0 0 0 T(3)11 T
(3)12 −1 0
0 0 0 0 T(3)21 T
(3)22 0 −1
0 0 0 0 0 0 −1 Zout
·
p1
u1
p2
u2
p3
u3
p4
u4
=
0
0
0
0
0
0
0
0
(37)
In equation 37, the transfer matrix elements are functions of complex frequency. By using a complex
eigenvalue solver the eigenfrequencies of the system can be determined numerically. For a system
without any sources or damping the imaginary part of the eigenvalues will be zero. If some small
amount of damping is present but still no source, any oscillations will decay in time and the imaginary
part of the eigenfrequency will be positive. However, by introducing a flame described by e.g. a time
20
lag model, an abrupt change of phase across the flame is introduced which constitutes a feedback loop
and unstable modes may appear. Furthermore, the corresponding eigenvectors represents the mode
shapes which can be used for further investigation of a systems dynamic behaviour.
2.6 Flame Models
Flames in network models are commonly represented by analytical expressions or measured transfer
functions. A large number of different flame models can be found in the literature. Within this
work, an analytical flame model is utilized. In recent years the rapid development of computational
performance has open for studying flames in reactive CFD models which has become an important
reference for tuning of analytical models, [1].
The most basic analytical models assumes the flame being acoustically compact. This assumption
is generally good in the low frequency range where the acoustic wavelength is much longer than the
flame region. For an acoustically compact flame the heat release distribution within the flame are less
important and can be neglected. However, in a temporal perspective the flame cannot be assumed as
compact as will be described later.
Within a flame the mechanisms going on involves different time lags. This can be understood by
the fact that a flame within a gas mixture propagates with a certain speed. Hence, a burning com-
bustible mixture need some time to react. In addition, the flame in premixed combustors is located
some distance downstream of where the fuel is injected and the fuel burnt in the flame at any instant
of time was injected at an earlier time. Obviously, disturbances created upstream the burner will
reach the flame after a certain time delay. The analytical models usually involve these time lags and
it is well known from control theory that systems involving time lags are inherently unstable.
2.6.1 Acoustic Jump Conditions
The so called jump conditions constitute the basis in many analytical flame models. The jump
conditions relates the acoustic pressure and velocity downstream of a compact flame to the upstream
state. For the derivation of the jump conditions, a thin flame with thickness ∆xf and volume Vf as
outlined in figure 12 is considered, and
21
Figure 12: Schematic illustration of a thin flame
the linearized acoustic conservation equations are integrated over the flame volume. Starting with
acoustic momentum, equation 26 gives
˚Vf
(ρ∂u′
∂t+∂p′
∂x
)dV = A
ˆ∆xf
(ρ∂u′
∂t
)dx+A
ˆdp′ = 0 (38)
In the limit of an infinitely thin flame, ∆xf approaches zero. Since the integrand in the first integral
being finite this integral will vanish and the result obtained is that the acoustic pressure is unchanged
over a thin flame.
p′downstream(t) = p′upstream(t) (39)
The same procedure is applied to equation 25, integration over the flame volume yields
˚Vf
(∂p′
∂t
)dV +
˚Vf
(ρc2
∂u′
∂x
)dV =
˚Vf
((γ − 1)q′) dV. (40)
In the limit of an infinitely thin flame the first integral vanishes and the equation simplifies to
A
ˆdu′ =
˚Vf
((γ − 1)
ρc2q′)dV. (41)
Since the total unsteady heat release rate is given by Q′ =˝
Vfq′dV , the relation for acoustic velocity
across the flame is obtained as
u′downstream(t) = u′upstream(t) +1
A
γ − 1
ρc2Q′(t). (42)
These results show the acoustic velocity features a jump across the flame in contrast to the acoustic
pressure which stays the same. The compact flame acts as a an acoustic monopole source creating a
22
fluctuating volume flow.
To complete the formulation of a flame model, an expression for the unsteady heat release rate (Q′(t))
is needed.
2.6.2 The n− τ Model
The classical way of modelling the unsteady heat rate release is the so called n − τ formulation
originally developed by L. Crocco in the 1950s for rocket engines. The n− τ model can be expressed
as, [23],
1
A
γ − 1
ρc2Q′(t) = n · u′upstream(t− τ). (43)
The unsteady heat release rate is assumed to be related to the velocity fluctuations upstream of the thin
flame. Furthermore, the heat release rate lags the velocity by the time lag τ and n is a proportionality
constant (often referred to as the interaction index) determining the degree of coupling of the flame
response to the velocity fluctuation. In the literature the model is often called ”the sensitive time lag
model” due to its characteristics being very sensitive to the time lag value. The frequency domain
formulation of the n− τ model is obtained by a Fourier transformation as
1
A
γ − 1
ρc2ˆQ = n · uupstreame−iωτ . (44)
In the above formulation, the acoustic velocity upstream of the flame is used as the reference for the
heat release. However, in the literature it is possible to find formulations were the heat release is
related to the acoustic velocity in some other location. How to estimate the time lag parameter will
be addressed in succeeding sections.
Combining equation 44 with the frequency domain version of equation 39 and 42 gives the acous-
tic jump condition for a thin flame to be
pdownstream = pupstream
udownstream = uupstream(1 + n · e−iωτ ).(45)
For stability analysis using complex eigenvalues, the above result is rewritten in the form of a transfer
matrix. The transfer matrix for a thin flame following the n− τ model reads
T =
(1 0
0 1 + ne−iωτ
). (46)
This is a passive flame model which means the acoustic state downstream of the flame does only
depend on the upstream state without any active source inside the element.
23
2.6.3 An Extended Flame Model
The classical n− τ model assumes the heat release fluctuations being related to the acoustic velocity
at a reference position. This is however not always sufficient, e.g. equivalence ratio fluctuations
originating from the pressure and velocity fluctuations at the fuel nozzle location will also influence
the flame response. In this section a more detailed flame model is given following a formulation
developed by Siemens and which is further described in [14] and [15]. The total heat release rate for
a combustion process is given by
Q(t) = yfρV hf , (47)
where yf is the mass fraction of fuel, V is the volume flow and hf is the specific reaction enthalpy.
By a series expansion around the mean and linearization, the expression for the unsteady heat release
rate is obtained as
Q′(t) = hf V yfρ′(t) + hfρyf V ′(t) + hfρV y
′f (t). (48)
The first two terms in equation 48 relates heat release to volume flow and density fluctuations just
upstream of the flame while the last term including y′f accounts for equivalence ratio fluctuations.
A simple burner featuring a sudden expansion as depicted in figure 13 is now considered. Fuel is
injected in the upstream part of the premixing section.
Figure 13: Schematic illustration of burner with fuel injection in the premixing passage.
The volume flow fluctuation (second term in equation 48) is assumed to be related to the acoustic
velocity fluctuation at a reference position with a delay (τb). This reference position is commonly set
to the burner exit plane and the time lag is given by the convective time from the burner outlet to
the flame. The volume flow fluctuation can then be expressed as, [14],
24
V ′ = u′b(t− τb)Ab, (49)
where the subscript b denotes the burner outlet. The volume flow fluctuations characterized by the
burner time lag is discussed by i.e. [19] and [23]. The underlying mechanism highlighted is the vortex
created at the burner outlet by the induced velocity fluctuations. This vortex is convected with the
mean flow through the flame region which makes the heat release to fluctuate. The approach to relate
volume flow fluctuations to the velocity at the burner outlet plane has been confirmed successful by
many studies on laboratory scale burners described by e.g. [5] and [19].
For gas turbine burners featuring turbulent and swirling flows, determination of the time lag is a
challenge on its own. Several different approaches has been suggested in the literature to determine
the time lag, i.e. [1] and [25] determined the burner time lag by fitting an analytical flame model to
measured flame response functions. W. Krebs et.al. [14] suggests a CFD approach to determine the
time lag.
An expression for the fluctuating mass fraction of fuel y′f can be obtained by studying the fuel mass
flow through a fuel nozzle. Assuming incompressible flow, the mass flow through the nozzle is given
by
mfuel = Anozzle
√2ρfuel(pfuelline − ppremix). (50)
where Anozzle is the effective area of the nozzle and pfuelline is the pressure of the fuel feed. The
pressure in the premixing section is given by ppremix and hence ∆pnozzle = (pfuelline − ppremix)
constitutes the pressure difference over the nozzle. The mass fraction of fuel can then be expressed as
yf =mfuel
mair + mfuel≈ mfuel
mair=Anozzle
√2ρfuel(pfuelline − ppremix)
ρpremixupremixApremix. (51)
Moreover, the fuel line feed pressure, pfuelline is assumed to be constant while the pressure, velocity
and density in the premixing passage are acoustic quantities. By a series expansion around the mean
and linearization, the fluctuating mass fraction of fuel can be expressed as, [14],
y′f (t)
yf= −
p′premix(t− τi)2(pfuelline − ppremix)
−u′premix(t− τi)
upremix−ρ′premix(t− τi)
ρpremix. (52)
Hence, the fluctuating mass fraction of fuel at the flame is dependent on pressure, velocity and density
fluctuations in the position of fuel injection. Disturbances generated are convected to the flame with
the mean flow. Furthermore, the fluctuating mass fraction of fuel at the flame front will be delayed
by the fuel time lag (τi). Upstream the flame, the isentropic relation holds and is now used to rewrite
25
the last term in equation 52 in terms of acoustic pressure instead of density. Putting it all together,
the unsteady heat release rate is found to be
Q′(t) =hf V yfp′flame(t)
c2flame+ hfρbyfu
′b(t− τb)Ab
+ hfρpremixV yf
(−
p′premix(t− τi)2(pfuelline − ppremix)
−u′premix(t− τi)
upremix−p′premix(t− τi)ρpremixc
2premix
) (53)
It should be mentioned here that in the premixing section, the strength of the equivalence ratio
fluctuations will reduce due to the mixing. This phenomena is not included in this modelling approach
and the equivalence ratio fluctuations are assumed to have the same strength when reaching the flame
as when they were produced.
2.6.4 Fuel Injection Time Lag τi
Fluctuations in equivalence ratio will be produced due fluctuations of the pressure drop over non-
choked nozzles which makes the fuel supply rate to oscillate. In addition, fluctuations of equivalence
ratio will also be created due to fluctuations of the air supply. Referring to figure 13 and considering
a pressure fluctuation at the flame. The events following due to this pressure fluctuation is illustrated
in figure 14.
Figure 14: Qualitative description of the contributing parts to the fuel injection time lag, from [18].
The time lag characterizing the unsteady heat release due to equivalence ratio fluctuations can be
divided in 3 parts. First part, there will be a phase difference between the pressure oscillation at
26
the nozzle and the flame due to the distance. The time lag associated with this phase difference is
denoted τ1. In the low frequency region and with a distance between fuel nozzles and the flame being
in the range of 100mm, this phase difference will be small and can in most cases be neglected. Second
part, the equivalence ratio fluctuations generated at one instant will be convected by the mean flow
reaching the flame at a later instant. The delay associated with this transport time is denoted τconv.
For simple burner configurations this τconv can be estimated by the mean air velocity and the length
between the fuel injector and the flame. However, burners used in gas turbines often comprises a
swirler generating turbulence which complicates the estimation of the convective time lag. In such
case, a detailed CFD analysis of the turbulent flow field can be used to determine the convective time
lag as suggested by [24]. Third part, when the equivalence ratio fluctuations reaches the flame the heat
release will occur some time later due to the delay related to the chemical reaction, here denoted τchem.
Moreover, the total time lag for equivalence ratio fluctuations is given by τtot = τ1 + τconv + τchem
when the acoutic pressure at the flame is used as reference. Following the definition used in equation
53, the fuel time lag is referenced to the acoustic perturbations at the fuel injection position and hence
the time lag will be given by τi = τconv + τchem. At low frequencies τchem/T will be small and the
time lag associated with the chemical reaction can in most cases be neglected, [19].
2.6.5 Distributed Flame Models
The thin flame models described in preceding sections are based on the assumption of acoustic com-
pactness. This assumption does not generally hold in practice. In the low frequency range the wave-
lengths will be much longer than the spatial distribution of the flame. Hence, in a spatial perspective
the flame may be considered as compact and the compact flame approach holds, [25]. This is however
not valid in a temporal perspective which can be illustrated as follows. It has been shown that the
time lags involved in stability analysis are dominated by convective transport times. Typical mean
flow velocities in gas turbine combustors are around 10 − 80m/s. Assuming the flame is distributed
over an axial distance of 100mm and the mean flow velocity being 50m/s gives the convective time
to pass the flame to be 2ms. At 200Hz the period of oscillation is 5ms and hence τconv/T is not
negligible and the flame cannot be assumed as compact in a temporal perspective. A more suitable
model should therefore include a distribution of the heat release as depicted in figure 15. The time
lag is here a function of x to describe the distribution within the flame.
27
Figure 15: Schematic illustration of burner with a distributed conical flame.
Many different flame models can be found in the literature featuring distributed heat release and time
lags. E.g. a flame model with distrubuted heat release is descibed by [29]. The local heat release rate
is then expressed as
q′(x, t) = n(x)q
uu′(xref , t− τ(x)) (54)
where τ(x) is the time lag distribution, xref is the reference location for velocity perturbations and
n(x) is an interaction index describing the spatial distribution of the heat release within the flame zone.
For acoustic characterization of a flame, another common used approach is to study the flame transfer
function (also denoted flame response function) defined as
F (ω) =ˆQ(ω)/Q
uref (ω)/uref. (55)
The flame transfer function relates the unsteady heat release rate to the acoustic velocity fluctuations
at a reference position. For easy comparisons, the reference position is normally selected at a loca-
tion where the acoustic velocity is easily measured. The heat release is commonly measured by OH*
chemiluminescence techniques. Measured flame response functions are commonly used to calibrate
an analytical flame model. This has e.g. been done by [25] who tuned an analytical flame model in
which a probability density function was used for description of the time lag distribution.
Distributed flame models is not the topic of this work.
2.7 Boundary Conditions at Combustor In- and Outlet
Determination of the up- and downstream acoustic boundary conditions are important for accurate
modelling of thermoacoustics. Poinsot, [16], showed that even with a detailed model of the combus-
tor, the result is still very dependent on the boundary conditions. However, these acoustic boundary
conditions are not easily determined for a gas turbine combustor at operation conditions. A detailed
28
review of possible boundary conditions for acoustic eigenmode calculations can be found in, [16].
A gas turbine compressor is used to increase the static pressure. The stationary guide vane stages in
an axial compressor therefore feature diverging flow channels to reduce velocity and increase static
pressure. For an acoustic model of a gas turbine combustor the upstream boundary is commonly set to
the outlet of the last compressor guide vane stage. In contrary, the flow out from the combustor flows
through the turbine guide vane nozzles which feature converging flow channels in order to increase
velocity. For acoustic modelling of a gas turbine combustor, the inlet to the first turbine guide vane
stage is commonly specified as the downstream boundary.
For low frequencies the acoustic boundaries can be approximated from the theory of converging
and diverging compact nozzles. Under this assumption some analytical expressions for the acous-
tic boundary can be found in the literature. In the derivation of such expressions it is assumed that
the characteristic length is small compared to the acoustic wavelength and hence geometrical details
can be neglected. Consider the flow in the first turbine guide vanes at the combustor outlet to be
choked converging nozzles. This chocked nozzle assumption implies acoustic waves can only travel in
one direction through the nozzle and hence no acoustic waves from the turbine section can travel up-
stream to the combustion chamber. For a choked converging nozzle with the upstream Mach number
(M1) being low, the reflection coefficient can be expressed as, [16],
R1 =1− 1
2 (γ − 1)M1
1 + 12 (γ − 1)M1
. (56)
Where the index 1 denotes the reflection coefficient for acoustic waves incident on the nozzle from
upstream. It should be noted that for low upstream Mach numbers, M1 → 0 and R→ 1, acoustically
the boundary acts as a hard wall.
Similarly, for a choked diverging nozzle the reflection coefficient can be expressed as a function of
the downstream Mach number (M2) as, [16],
R2 =1− γM2 + (γ − 1)M2
2
1 + γM2 + (γ − 1)M22
. (57)
Here, index 2 denotes the reflection coefficient for waves indent from downstream on the nozzle.
Moreover, if the downstream Mach number being low (M1 → 0), the diverging nozzle will also act an
acoustically hard wall (R→ 1).
For non-choked nozzles, acoustic waves can travel in both directions. Analytical expressions of the
reflection coefficient for non-choked compact nozzles can be established for both diverging and con-
verging geometries, [16]. However, those requires information of the acoustic boundary at the troat of
the nozzle to be known. Analytical expressions for non-choked nozzles and suggestions for the troat
29
boundary conditon are given in [16]. Commonly for the compressor outlet vane row, constant flow
velocity rate is assumed at the inlet to the guide vanes which means u′ = 0 and Rvane inlet = 1. For
turbine nozzles, constant pressure may be assumed at the outlet of the guide vanes and hence p′ = 0
and Rvane outlet = −1.
For higher frequencies the detailed geometry of the inlet and outlet vanes cannot longer be neglected.
The reflection coefficient can then be solved for in a quasi-one-dimensional manner by linearizing the
Euler equations. The result will be a frequency dependent boundary conditon. This approach is fur-
ther described by Poinsot, [16]. Another extensive study of the downstream boundary condition using
LES can be found in [8]. Some authors have also tried measuring the reflection coefficients for the
inlet respective outlet, i.e. [27] measured the reflection coefficient at the combustor inlet and outlet
for a model gas turbine.
30
3 Method and Numerical Tool
3.1 Network Modelling Tool
In the remainder of the report, a generalized network code developed by Siemens AG is used to
study thermoacoustic stability as outlined in the theory section. The network code solves stationary
flow, kinetics, heat transfer, thermodynamics and acoustics in a coupled manner. This means that
an acoustic model must also include relevant flow features and fuel spices to capture heat release,
temperatures etc. which are of importance for thermoacoustics. A built in library with predefined
elements is available and the code features a graphical interface through Simulink in which the elements
and interconnection of elements are defined. MATLAB is used as the primary solver for the underlying
equations and the open source package Cantera has been integrated for solving kinetics and chemical
reactions. Two types of acoustic analysis is available, these are forced response analysis and complex
eigenfrequency analysis. The thermoacoustic module has previously been validated by Siemens AG
against an academic burner with successful results.
3.2 Sample Cases
Several sample cases were created and investigated to learn the code and understand the behaviour
of the available acoustic elements. Some of those sample cases are described in this section.
3.2.1 Eigenfrequencies for a Simple Duct
As a first study, the eigenfrequencies for a simple duct with hard walls at both ends were studied and
the results were compared to well known analytical expressions. The available complex eigenvalue
solver was used and the network model for the simple duct is shown in figure 16. The working
procedure for the solver is as follows. First, the flow and chemical reactions are solved and when
finished the acoustic solver is initiated. Input parameters needed for the acoustic study such as speed
o sound, Mach number etc. for each node in the model are automatically imported from the flow
simulation.
Figure 16: Network model of a simple duct.
Air was used as the medium and a parameter study was performed with different mass flows to capture
the effect of a mean flow on the acoustics. The analysis was done for a duct with length 1m and a
total temperature of the inlet air being 293K. The eigenfrequencies determined by the network code
as a function of the Mach number is shown in figure 17 along with the analytical eigenfrequency for
the n:th mode given by
31
fn =nc
2L(1−M2). (58)
The results from the complex eigenvalue solver correspond well to the analytically obtained eigenfre-
quencies. Slight differences are seen at higher Mach numbers which is a result of the coupled modelling
approach utilized in the network code. The analytic eigenfrequencies are here calculated by assuming
a constant speed of sound while for the network model, total conditions are specified and the static
conditions and thereby the speed of sound is recalculated depending on the mass flow rate. The
imaginary part of the eigenfrequencies was found to be almost zero as it should for a system without
sources and damping.
Figure 17: Eigenfrequencies for a simple duct as a function of the mean flow.
3.2.2 Wave Transmission through an Expansion Chamber
In the next study, wave transmission through an expansion chamber was studied using the forced
response analysis option. Within the code, the forced response analysis option splits the network at a
selected node for excitation. At this node a predefined acoustic velocity (u) is then enforced and the
acoustics is solved for each frequency in a specified range.
The dimensions of the expansion chamber used for this study is shown in figure 18. Air with a
temperature of 293K and without any mean flow was used as medium. The upstream boundary
was specified as the excitation node and the downstream boundary was specified as a reflection free
termination. The network model is shown in figure 19.
32
Figure 18: Illustration with dimensions of the expansion chamber.
Figure 19: Network model of the expansion chamber.
The transmission of waves incident from upstream was evaluated as the ratio of the incident pressure
amplitude (pi) to the transmitted amplitude (pt). The results was then compared to an analytical
expression found in [3]. Due to the modelling approach utilized for the forced response analysis,
decomposition of the acoustic field upstream of the expansion chamber was required to distinguish
the incident from the reflected wave. For decomposition the measurement method known as the two
microphone method was applied to the numerical results, the method is further described in Appendix
A. The results from the network model was found to correspond well to the analytic results as shown
in figure 20.
33
Figure 20: Incident to transmitted amplitude ratio for the expansion chamber.
3.2.3 Rijke Tube
As a first model including a flame, a network model of a simple Rijke tube was created and is shown in
figure 21. Up- and downstream boundary conditions was set to R = −1, i.e. open to atmosphere and
constant pressure. Within the network code, the acoustic flame model is merged into the available
reactor elements. A combustible air-methane mixture with temperature 300K, equivalence ratio
Φ = 0.67 and a mass flow of 1g/s was specified at the inlet. This mass flow gives a maximum Mach
number of 0.02 in the tube and the mean flow effects will be negligible. The flame divides the tube
into a low temperature and a high temperature region. The temperature upstream of the flame is
given by the temperature of the inlet mixture while the temperature downstream is calculated by the
reactor for the given combustible mixture.
Figure 21: Model for thermoacoustic analysis of a Rijke tube.
Several types of chemical reactors are available in the network element library and the choice of re-
actor is strongly connected to the purpose of the model. For thermoacoustic analysis, all the reactor
elements use the same thin flame model including volume flow fluctuations and eqvivlence ratio fluc-
34
tuation and which is futher described in [15]. The coupling between the chemical reaction and the
acoustics is the heat release in the flame. An equilibrium reactor assumes infinite reaction time in-
side the reactor and the temperature downstream will then be the adiabatic flame temperature. The
equilibrium reactor was selected to be used through out this work.
For the Rijke tube, no equivalence ratio fluctuations were included as a homogeneous air-fuel mixture
was specified at the inlet. The flame model will therefore have the form of a simple time lag model.
Results from the complex eigenfrequency solver is shown as a stability plot in figure 22. Positive
values indicates unstable modes that may grow in time.
Figure 22: Thermoacoustic stability plot for the Rijke tube.
Complex eigenfrequencies were calculated for three cases. First case with only a tube with two dif-
ferent temperature regions and no flame (i.e. no jump in acoustic velocity at the interface of the two
temperature regions). The second and the third case comprise a flame with a time lag being τ = 0
and τ = 0.1ms. Furthermore, the Rijke tube has no sudden expansion as in the derivation of the
flame model and volume flow fluctuations were related to the immediate upstream side of the flame.
Interpretation of the time lag for a Rijke tube where the flow could be assumed to be close to laminar is
therefore a bit tricky. Since no obstacles generating disturbances or such thing, the convective time lag
assumption cannot be applied and the time lag will be more connected to the chemical reaction. Any-
how, the actual value of the time lag was not analysed in depth for the Rijke tube but will be further
investigated in succeeding sections. It was the behaviour of the flame that was of interest in this study.
Conclusions
• Zero time lag (τ = 0) implies an acoustic velocity jump at the flame but no change of phase.
This does not create any unstable modes but a significant change of the eigenfrequencies are seen
compared to the no flame case (no velocity jump). The reason to the change in eigenfrequencies
35
is due to the changed impedance at the interface between the hot and cold temperature regions.
The system will still be stable due to if no time delay the acoustic pressure and unsteady heat
release will be 90 degrees out-of-phase and the Rayleigh’s integral will render zero.
• Introducing a time delay will make some modes to become unstable and some to be damped
depending on the phase between the acoustic pressure and the unsteady heat release. For this
Rijke tube configuration, unstable modes are predicted at around 190Hz and 680Hz.
• Positioning the flame in the downstream half of the tube was found to not create any unstable
modes as expected (stability plot for this study is not shown here).
3.2.4 Simple Burner Featuring a Sudden Expansion
As a next sample study, a network model of a simple burner featuring a sudden expansion with area
ratio=2 (A2/A1) was created. The model with the baseline dimensions and input parameters is shown
in figure 23. Dimensions were arbitrarily selected without any further reference. Acoustically the in-
let boundary was specified as a hard wall while outlet was assumed to be an open end. As for the
Rijke tube an equilibrium reactor was used to include the flame. As a first case, a perfectly premixed
combustible air-methane mixture with equivalence ratio Φ = 0.67 and a mass flow of 0.1kg/s was
specified at the inlet. The mass flow of 0.1kg/s gives a maximum Mach number of 0.17 at the outlet.
Figure 23: Sample burner model. Baseline case.
The heat release fluctuations at the flame are assumed to be related to volume flow fluctuations at
the burner exit plane only (as introduced in equation 49). The burner time lag (τb) is dominated
by the convective time a fluid particle need to travel from the sudden expansion to the flame. It is
obliviously crucial to determine the position of the flame in order to determine the burner time lag.
A parameter study was performed varying the input parameters and geometrical dimensions up and
36
down from the baseline case. Only one parameter was varied at a time. Stability plots for different
input parameters and geometrical dimensions are shown in figure 24. The stability measure on the
y-axis is here presented as the growth rate defined as
Growth rate = exp
(−2π
Im(f)
Re(f)
)− 1. (59)
The growth rate is a measure of how much the amplitude is changed over one cycle. For a positive
growth rate the oscillation grows and for a negative growth rate the oscillation is damped. A growth
rate of unity means the amplitude will increase by 100% (doubling) over one period of oscillation.
For a high growth rate it is easily understood the amplitude may grow very fast and give rise to
instabilities. However, since this is a linear code the growth rate cannot be used to determined the
final limit-cycle amplitude which is controlled by non-linear phenomena.
37
Figure 24: Stability plots. Sensitivity study to different input parameters and geometrical dimensions.
Conclusions
• Geometrical changes such as varying L1 and L3 will change the acoustic response of the system
and the geometry itself is hence strongly connected to stability. Making L1 shorter seems to
be better for this configuration and it is possible to find a value of L1 for which no unstable
modes at all is predicted. For the second mode, the length of L1 has a significant influence
on the frequency as well. Also increasing the length of the outlet pipe (L3) is predicted to act
38
stabilizing for this burner configuration. The results shown here indicate very high sensitivity
to geometrical changes. However, it should be said that the geometrical changes investigated
here have been quite dramatic compared to the size of the model.
• The position of the flame (L2) has a minor influence on stability given a fixed value of the time
lag. This follows from the frequently good assumption of the flame being acoustically compact
in a spatial perspective. At higher frequency it can be seen some change in frequencies due to
the change of flame position is larger with respect to the wavelength.
• Equivalence ratio influences the growth rate slightly but not so much the frequencies. This is a
bit counter intuitive since a higher equivalence ratio (still below 1) will give a higher downstream
temperature and hence the eigenfrequencies should go up. However, the unsteady heat release
will also go up and those phenomena together will determine the response.
• The time lag and hence the phase change of velocity across the flame has a large influence on the
frequencies but not so much the growth rate. Hence, the flow velocity controls the convective
time lag and will be very important even though the influence on the acoustic field is small for
low Mach numbers.
The burner was now modified to include fuel injection nozzles to study stability due to equivalence
ratio fluctuations. In the flame model implemented in the network code, the expression for equiva-
lence ratio fluctuations introduced in equation 52 is slightly rewritten in terms of fuel line impedance,
this is futher described in [15]. The fuel line impedance characterizes the fuel line and is defined as
the ratio of acoustic pressure to acoustic velocity at the outlet of the fuel injection holes. Since the
rate of fuel injection is determined by the velocity through the nozzles, a high fuel line impedance
value means the velocity fluctuations will be small and the rate of fuel injected becomes more or less
constant. Anyway, with high fuel line impedance values implying a constant fuel supply, equivalence
ratio fluctuations will still be present due to fluctuations of the air in the pre-mixer.
Two fuel injection locations were specified as outlined in figure 25. In the network code, this is
done in a merger element where the air and fuel are mixed. Within the merger element it has to
be specified that equivalence ratio fluctuations are to be included in the thermoacoustic study. This
makes the acoustic perturbations in the location of fuel injection to be used as reference for equiva-
lence ratio fluctuations. The same amount of fuel as in before was specified in order to give the same
overall equivalence ratio at the flame and hence the same downstream temperature. Within this study
a very high value of the fuel line impedance was used which was justified by a large area difference
between the fuel injection nozzles and the burner premixing section. The influence due to volume flow
fluctuations at the burner exit plane was excluded in this study in order to investigate the influence
of equivalence ratio fluctuations only.
39
Figure 25: Sample burner model, with two locations for fuel injection added.
As a first case, the importance of the location of the fuel injector was studied by letting all the fuel
to be injected through the mfuel,1 port. Stability plots for different injector locations as well as
different fuel time lags are shown in figure 26. For equivalence ratio fluctuations the fuel time lag
(τi) is dominated by the time required for a fluid particle to travel from the fuel injection location
to the flame. The results confirm the location of the injector has a small influence on the stability
for a fixed fuel time lag. This is due to the long wavelength and hence the phase difference between
the different injection locations is very small. The value of the fuel time lag on the other hand has a
large influence on both stability and frequencies which is in line with previously observed results for
volume flow fluctuations (controlled by the burner time lag).
Figure 26: Stability plots. Influence due to equivalence ratio fluctuations.
The fuel was now split equally between the two fuel injection locations as shown in figure 25 in order
to investigate the interaction between two fuel injectors. The fuel time lag for the injector at location
1 was kept constant while the fuel time lag for the injector in location 2 was varied.
40
Figure 27: Stability plot. Interaction of two fuel injection positions.
Its clearly seen for the first mode that increasing the fuel time lag for one of the fuel nozzles is pre-
dicted to act stabilizing. This is however only true to a certain limit which can be understood from
the period of oscillation. The criteria for instability is given by the Rayleigh’s integral and includes
the phase of the pressure and the unsteady heat release. More in detail, the integral will change sign if
the time lag is lager than T/2 (but still less than T ). It is often useful to think of time lags in relation
to the period of oscillation in order to get an idea of the expected influence to a certain change of
time lag. For this case, the first mode has a period of oscillation of T = 5ms (1/200). Hence, a time
lag difference of 2ms as is the maximum change investigated here is significant and a large impact is
expected on stability. For the higher mode at around 400Hz the period of oscillation is 2.5ms and
half of that is 1.25ms. Hence a time lag difference of 2ms are expected to have a large impact on
stability but may also flip the results around totally. For this configuration τi,2 = 4ms seems to be
better for stability than τi,2 = 5ms. Since there are several different terms included in the full flame
model, the response to a certain change of time lag is anyhow difficult to predict which is why the
model becomes very useful. However, the results suggest that a careful selection of the time lags could
be an efficient way to improve stability.
In this section, some sample cases have been studied and the results have been discussed. It has
been shown that even for simple models the stability behaviour becomes complicated and difficult to
predict before hand. Parameter studies was found to be very useful to investigate trends even though
too many cases may be more confusing than helpful. In succeeding sections, the network modelling
approach will be applied to a commercial gas turbine combustor.
41
4 Application to the SGT-750
The Siemens SGT-750 has a combustion system consisting of eight identical cans. At present, the
combustion system provides excellent combustion stability and performance. However, a deeper knowl-
edge in its thermoacoustic response will be useful for possible future upgrades and further reduction
of emission levels. The network modelling approach as outlined in the previous sections was therefore
applied to the SGT-750 and the results were compared to measurement data.
Considering low frequencies and a single can, no transverse modes can exist due to the relatively
small cross-sections. With the largest diameter, the cut-on frequency for the first transverse mode is
of the order of 1000Hz for the SGT-750 combustor at operating conditions. In a full engine though,
longitudinal modes are not the only ones that can exist at low frequencies.
Exhaust gases from the combustion chamber are led through a transition duct to the turbine inlet.
Due to thermal expansion reasons the transition duct is not tightly sealed to the turbine guide vanes
and a gap between the transition duct and the guide vanes is required. This gap provides a connection
for the different cans to acoustically interact at the hot side. Additionally, the air feed from the com-
pressor diffuser to each can is through a common casing which creates the possibility for the cans to
acoustically interact at the cold side as well. By this said, the conclusion is that there is a possibility of
can-to-can interaction which may be present at low frequencies together with longitudinal oscillations.
The thermoacoustic stability study for the SGT-750 was started with an investigation of available
measurement data after which the network modelling approach was applied and the results compared.
4.1 Evaluation of Measurement Data
Available measurement data from a prototype engine test was analysed. For the SGT-750, combustion
dynamic levels are measured in the upstream part of each burner using transducers with high tem-
perature resistance. Two different full load operation conditions were investigated, SGT-750 standard
operation conditions and conditions when instability was provoked. The measurement data was avail-
able as time signals which were processed using the MATLAB signal processing package. Auto power
spectrum for the two conditions are shown in figure 28, the frequency axis has been normalized and
the same normalization is used from now on for easy comparisons. As expected, the dynamic levels
are much higher for all cans when instability is provoked even though there are quite large differences
between the individual cans. It can be observed the dynamic response is dominated by two distinct
frequencies for both operation conditions. Those two peaks are from now on referred to as the first
respective second peak where the first occurs at a lower frequency than the second.
42
Figure 28: Auto power spectrum. Left figure: Standard operation. Right figure: Provoked instability.
To investigate if the signals from the individual cans were correlated, the coherence measure was
calculated. The coherence between two signals x and y is defined as
Cxy =PxyP
∗xy
PxxPyy, (60)
where Pxx and Pyy is the auto spectrum of the respective signal and Pxy the cross spectrum between
the two. Coherence was calculated with each of the cans as the reference for the provoked instability
condition. As an example, the coherence with burner/can no.5 as the reference is shown to the left in
figure 29.
Figure 29: Left figure: Coherence for each can with can no.5 as reference. Right figure: Transfer
function phase between each can and can no.5.
In general, low coherence values were achieved which indicate the signals are not coherent or high
43
levels of noise. However, at the frequency of the first peak seen in the auto power spectrum, the
coherence shows a peak as well indicating it could be some correlation. For each can as reference, the
distinct peak in coherence was observed for the other cans except two. In figure 29, with can no.5 as
reference it is seen the coherence is low for can no.3 and no.7 at the frequency of the first peak. At
the frequency of the second peak though, the coherence is about zero and hence no correlation at all
between the individual cans. Furthermore, if two signals are coherent it is of interest to determine
the phase difference between them. The phase difference between each can and the reference was
investigated by studying the phase of the cross spectrum since it is the same as the transfer function
which can be estimated as
Hxy =PxyPxx
. (61)
The phase of the transfer function between each can and can no.5 is shown to the right in figure 29
for the interesting frequency range. A rectangular window was used in the data processing in order
to retain the phase as accurate as possible and enough data was available to perform a large number
of averages. Referring to figure 29, the phase shown is the phase difference to the reference can no.5.
The phase for can no.3 and no.7 should not be considered since the coherence indicated those to be
completely uncorrelated to the reference. For the other cans, it can be seen that the phase difference
for can no.4 and no.6 is close to zero indicating they are in-phase. The remaining cans has a phase
difference of about ±π which indicates they are completely out-of-phase to the reference. This means
the reference can is in-phase with the closest cans while being out-of-phase with the cans on the
opposite side of the engine. The two cans with low coherence was found to form a line in between
the in-phase and out-of-phase side. That explains the low coherence which is probably due to a zero
crossing and thereby low signal levels. Similar results where obtained independent on the selection of
reference can. The results are visualized in figure 30.
44
Figure 30: Phase relations at the frequency of the first peak.
The results show the first peak seems to a global mode including all of the cans. This global mode
has also been found in previous FEM-calculations of the full SGT-750 combustion system. By this
finding, it is apparent that this instability mode cannot be predicted if only longitudinal modes in one
combustor can are studied.
Conclusions
• There are two distinct frequencies that dominate the thermoacoustic response at low frequencies.
• Coherence indicates the first peak (lower frequency) most likely is coherent between the cans
and hence characterized by a global mode including all of the cans.
• Further study of the transfer function phase showed this probably is a mode with nodal diameter
1 where each can is in-phase with the two closest cans and out-of-phase with the three cans on
the opposite side of the engine.
• Low coherence at the frequency of the second peak indicates no correlation between cans. This
frequency is due to a local phenomena in the individual cans.
4.2 Stability Analysis by the Network Modelling Approach
4.2.1 Establishing the Acoustic Network Model
An acoustic network model of a 45 degree sector was developed to be used for stability analysis of the
second peak. The model comprises one of the identical cans and the compressor diffuser and central
45
casing featuring annular cavities was divided accordingly. The objective of the model was to capture
the low frequency behaviour and the modelling strategy was therefore to:
• Capture lengths and over all volumes.
• Capture effective area and flow velocities in the fuel injection passages (velocities in premixing
section are important in the study of equivalence ratio fluctuations)
The geometry of the combustor system was divided into duct, area discontinuity and variable area
section elements. Dead volumes of significant size was modelled as side branches to capture quarter
wave resonator effects. Some significant assumptions had to be made for modelling of the common
casing since the geometry was difficult to translate into a 1-D model. The main focus for the casing
was to capture the volume and cross-sectional area. This since the casing volume is significantly larger
than for the swirlers and will be important for correct acoustic reflections at the interfaces. The split
into different elements is shown in figure 31.
Figure 31: Schematic illustration of the network model. Division into acoustic network elements.
The SGT-750 burner comprises two radial swirlers in where the main fuel is introduced. Due to low
Mach numbers, the radial swirlers were assumed to be acoustically transparent meaning any flow
effects generated by the swirlers were neglected in the acoustic model. Each of the passages in the
swirler were modelled by duct elements with specified effective area in order to get the air split between
the two swirlers correct. The final version of the network model is shown in figure 32. Sub-models for
the swirlers and flame region were created which will be further described in succeeding sections.
46
Figure 32: The acoustic network model.
The geometrical dimension of the model were specified directly in each element while the inlet param-
eters such as compressor mass flow, inlet pressure and temperature were specified in an excel sheet
for easy adjustments to different operation conditions. The total mass flow from the compressor was
divided by eight to get the mass flow through one of the cans. Cooling air consumed by downstream
parts of the combustor, mainly the transition duct and the interface between combustor and turbine
was extracted from the casing. This was required to get the correct amount of air in the combustion
zone and since a coupled solver, the amount of air in the combustion zone together with the amount
of fuel will determine the temperature in the flame.
Boundaries at both the inlet and the outlet were specified as acoustically hard walls for now. This
assumption will be further investigated in the next section.
Referring to figure 31, a stream wise coordinate from the combustor inlet to the outlet of the model
was defined. This stream wise coordinate is used for plots of acoustic mode shapes in succeeding
sections. The scale of the stream wise coordinate is element wise and hence not actual lengths which
should be kept in mind. Anyway, this representation of the mode shape is very useful to understand
the systems behaviour. Following Rayleigh, the location of the flame within a mode is of high interest
and the location of the flame is therefore indicated in the mode shape plots.
The eigenfrequencies and corresponding mode shapes from the network model were compared against
available FEM calculations for the case with a uniform temperature and pressure corresponding to
the compressor discharge air at full load. The mode shapes from the network model are shown in
figure 33 and the longitudinal mode shapes from FEM calculations are shown in figure 34. It can be
seen the mode shapes corresponds well. However, the frequencies was found to differ by up to 10%.
47
Figure 33: Acoustic mode shapes from the network model.
Figure 34: Acoustic mode shapes from FEM calculations.
4.2.2 The Influence of a Mean Flow on the Acoustics
The presence of a mean flow makes the speed of acoustic wave propagation in the upstream respective
downstream direction to be different. For gas turbine combustors in general, the Mach numbers are
low due to the high temperatures and hence high sound speed. The maximum Mach number within
the SGT-750 combustor is about 0.2. Higher Mach number values are obtained outside the acoustic
domain in the turbine guide vanes which constitutes the acoustic boundary at the combustor outlet.
The influence of the mean flow on the acoustics was investigated by comparing the eigenfrequencies
obtained by the network model with and without a mean flow. The resulting eigenfrequencies is
presented in table 1.
Full load conditions No mean flow
0.23 0.23
0.60 0.60
0.78 0.79
0.97 0.98
Table 1: Normalized acoustic eigenfrequenices with and without a mean flow.
As can be seen the differences in eigenfrequencies due to the mean flow was found to be very small.
48
A slight difference is seen for higher frequencies while the lower is barely changed. The conclusion
is that a correct mean flow is not crucial for the acoustic analysis. Anyway, the mass flow and flow
velocities will be of importance later on when convective time lags are to be estimated.
4.2.3 Introducing the Flame - Perfectly Premixed Case
In this section, the flame is introduced and the stability due to volume flow fluctuations at the flame
is studied. No equivalence ratio fluctuations are included yet.
Results from a previously performed CFD calculation was used to estimate the position of the flame.
The actual flame has a conical shape with significant axial distribution while an infinitely thin flame is
assumed in the flame model available. To tackle this mismatch, the axial position that best represents
the heat release was estimated to be in the middle of the actual flame. This position was found to
be just before the combustion chamber expansion section. The thin flame together with the actual
conical flame are illustrated in 35.
Figure 35: Schematic illustration of the network model. Including the flame.
The flame was introduced as an equilibrium reactor element further described in the study of the
Rijke tube. The sub-model for the flame region is shown in figure 36.
Figure 36: Network sub-model of the flame region.
49
Pure methane was used as fuel and an amount corresponding to full load conditions was introduced at
the inlet to the model as depicted in 35. The temperature downstream of the flame is determined by
the chemical reactor and was checked to correspond to the known flame temperature of the SGT-750
at full load.
Volume flow fluctuations at the flame are related to a reference position which for the sample burner
was set to the burner outlet plane (sudden expansion). This burner outlet plane was selected with
the argument that a vortex will be shed at this locations and is then convected to the flame. For the
SGT-750 burner there are no such distinct burner outlet plane. Instead, the volume flow fluctuations
were referenced to the end of the split plate which is where the main 1 and main 2 swirl flows will
meet and flow disturbances may be induced.
The volume flow fluctuations at the flame will lag the velocity fluctuations at the reference loca-
tion and the time lag had to be explicitly specified. This burner time lag consist of a convective part
and a chemical part as outlined in the theory section. The convective part of the burner time lag
was estimated from the bulk flow velocity and axial distance. In general the convective part of the
time lag is dominating but by this assumption of reference plane and flame location, the resulting
time lag is very short. This means the time lag due to the chemical reaction may not be negligible in
comparison to the convective time lag. However, both time lag components will be small compared to
the period of oscillation at low frequencies and hence the exact value should not be very important.
To investigate the sensitivity to the burner time lag parameter the stability for the full combustor was
calculated for some reasonable values of the burner time lag. The result is shown in figure 37.
Figure 37: Stability plot for different values of the burner time lag.
An unstable mode is predicted at a frequency corresponding to the second peak seen in the measure-
50
ment data evaluation. Varying the estimated burner time lag slightly up and down results in slightly
different frequencies but more or less the same growth rate. Hence, for this modelling approach under
the assumptions described above the approximation of the burner time lag seems to be reasonable.
Another mode is found at a normalized frequency of around 0,6 which for some time lag values has
positive growth rate and hence may be unstable. This originates in the period of oscillation for this
higher frequency is shorter and hence a smaller time lag change is required to make a stable mode
unstable and vice versa. No acoustic damping is included in the model and the growth rate only gives
an indication of frequencies that may become unstable.
Following the theory, the spatial location of the flame should not be crucial for low frequencies if
the proper time lag is used. This was confirmed by the study of the sample burner and a trial with
the flame located downstream the expansion was performed for the full model as well for a fixed
time lag, the differences were found to be very small. Anyway, the modelling should reflect what is
modelled as close as possible and a flame located upstream of the expansion as depicted in figure 35
was used for all succeeding studies.
According to the Rayleigh’s criteria an unstable mode will grow if the pressure is in-phase with
the unsteady heat release. This means that if the acoustic pressure is low at the location of the heat
release the instability is more difficult to trigger. The amplitude of the first two mode shapes are
shown in figure 38. For the first mode it is clearly seen that the acoustic pressure amplitude at the
location of the flame are high. Hence, the mode is easily triggered. On the other hand, for the second
mode the position of the flame is at or close to a node implying the mode is not that easy to trigger.
Figure 38: Mode shapes for different values of the burner time lag.
51
4.2.4 The Outlet Acoustic Boundary Condition
At full load, the flow velocity at the throat of the turbine inlet guide vanes are close to sonic. As
outlined in the theory section, the turbine inlet is for that reason commonly assumed to be choked.
Furthermore, the rapid acceleration of the exhaust gases takes place in a distance of a few centimetres
within the inlet guide vanes and the upstream Mach number in the combustor is still low. By this
arguments the reflection coefficient can be estimated from the theory of chocked compact nozzles
(equation 56) which gives a value close to R=1. However, since the network code allows for modelling
of variable area sections, a deeper study of the acoustic boundary was performed.
A small model was created with the inlet guide vane passage modelled as an area contraction section
shown to the right in figure 39. One guide vane passage was modelled and dimensions according to
the actual geometry was used. The throat of the guide vane passage is located as shown to the left in
figure 39, i.e. close to the trailing edge of the vanes. Hence, after the throat, the hot gases exhausts
into a relatively larger cavity in between the stator and rotor blades. By that reason the acoustic
pressure (p′) was assumed to be zero at the outlet of the contraction (vane) as suggested by [16].
Figure 39: Left figure: Schematic illustration of a turbine stage. Right figure: Model used for
estimation of the acoustic boundary.
A forced response study was performed for the interesting frequency range and the reflection coefficient
at the inlet to the contraction section R = pr/pi was evaluated, the result is shown in figure 40. It was
found that the reflection coefficient approaches 1 when the Mach number increases. The reason to the
flat behaviour of the reflection coefficient was found to be due to the acceleration in the guide vane
takes place in a very short distance compared to the wavelength. At higher frequencies or running
the simulation with an axially longer contraction section, the reflection coefficient will not longer be
independent of frequency. This is in line with the study of the turbine inlet boundary condition
performed by [8]. Only the amplitude of the reflection coefficient is shown here, the imaginary part at
those low frequencies was found to be negligible and hence the phase is not changed at the boundary.
52
Figure 40: Reflection coefficient amplitude for different Mach numbers at the throat.
A sensitivity study on the full SGT-750 model for different reflection coefficients at the outlet was
also performed. All other parameters were kept unchanged in this study. The resulting stability plot
is shown in figure 41 and the mode shape for the first mode in figure 42.
Figure 41: Stability plot for different values of the outlet reflection coefficient.
It was observed that the growth rate for the first mode significantly decreases with a lower reflection
coefficient while the frequency is more or less the same. The reason to the decrease in growth rate is
due to more acoustic energy leaves the system for lower values of the reflection coefficient. Also the
53
mode shape was found to be barely changed for the different values of reflection coefficient investigated.
Figure 42: Mode shape for different values of the outlet reflection coefficient.
Conclusions
For low frequencies at higher loads when sonic speed is approached in the inlet guide vanes, the hard
wall assumption seems to be a good first approximation. The main objective with network models
like this is to find trends, not exact values. Therefore, the reflection coefficient was be set to R=1
through out the succeeding studies.
4.2.5 Fuel Line Impedance
To estimate the fuel line impedance, separate network models were created for both main 1 and main
2 fuel supply systems. The principle for main 1 respective main 2 fuel lines is the same. The gen-
eral design of the fuel line is as follows. Fuel is introduced in the flow passages in the radial swirler
through numerous holes in order to achieve good mixing performance and a favourable fuel profile at
the swirler outlet. In the burner there are internal distribution channels in order to distribute the
fuel to the different injection locations. Outside the engine, each burner is connected by a connection
pipe to a large ring-like manifold which supply fuel to all burners. Furthermore, fuel is supplied to
the large ring-like manifold through a network of pipes and valves inside the gas turbine package.
A schematic view of a fuel line is shown in figure 43 and the network model for the main 2 fuel
line is shown in figure 44. Dimensions of the internal cavities were estimated as accurate as possi-
ble following available CAD-models. The upstream end of the model was set to the connection of
the connection pipe to the ring-like manifold outside the engine. Acoustic pressure was set to zero
(p′ = 0) at this surface since the cross-section of the connection pipe is relatively small compared to
the manifold. Flow parameters and temperatures were specified to correspond to SGT-750 full load
conditions and the impedance were extracted from a forced response analysis.
54
Figure 43: Schematic illustration of a fuel line.
Figure 44: Network model of the main 2 fuel line.
The length of the connection pipe for each burner to the manifold is slightly different for different
burners and varies between 0.9 to 1.2m. This will affect the resonance frequencies in the fuel supply
system and the fuel line impedance was therefore calculated for different lengths of the connection
pipe. The calculated fuel line impedances are shown in figure 45 and 46.
55
Figure 45: Fuel line impedance. Main 1 fuel line.
Figure 46: Fuel line impedance. Main 2 fuel line.
The phase of the fuel line impedances was found to slightly increase with frequency but still being
small in the frequency range of interest. The resonances of the fuel systems were clearly seen for
higher frequencies. Due to the fuel injections holes being small in comparison to the premixing cross-
section, the plane wave impedance amplitude was found to be large. This means the influence of a
fluctuating pressure drop over the nozzles will give small fluctuations in velocity within the nozzle.
Equivalence ratio fluctuations for this configuration will therefore be dominated by the fluctuations
of the air supply rather than modulation of the fuel flow rate. The estimated frequency dependent
fuel line impedances were used through out the following studies.
56
4.2.6 Equivalence Ratio Fluctuations
Thermoacoustic stability due to equivalence ratio fluctuations is now studied. The approximate lo-
cations of fuel injection for main 1 respective main 2 are shown in figure 47. There is also a fuel
injection location referred to as c-stage which is not there today but will be addressed and explained
in succeeding sections. The same amount of fuel as was used before was now introduced in the main
swirlers with a split between main 1 and main 2 corresponding to SGT-750 standard operation. Using
the same amount of fuel gives the same overall equivalence ratio and hence the same temperature
downstream of the flame. Fuel injection for each swirler passage was assumed to be through one
single injection hole with an effective area corresponding to the total effective area of the injection
holes present in one swirler passage. In the real burner, the spatial distribution of the small injection
holes will give a slight distribution of the convective time lag values. This effect was not captured
and only an average fuel time lag for main 1 respective main 2 were specified in the model. However,
the velocity of the fuel will still correspond to the actual value which is of relevance since the acoustic
velocity fluctuations in relation to the steady mean velocity determine the strength of the equivalence
ratio fluctuations.
Figure 47: Schematic illustration of the network model. Fuel injection locations.
The influence from volume flow fluctuations studied in section 4.2.3 was excluded in the flame model
for this study. A sensitivity study was performed for reasonable values of the fuel time lags estimated
from convective transport times. The result is shown in figure 48. Convective time lag values extracted
from CFD are used in next section.
57
Figure 48: Stability plot for different values of the fuel time lag.
Conclusions
• An unstable mode is predicted at the frequency of the second peak seen in the measurements
for all reasonable fuel time lag values.
• For some cases, a higher mode (at around normalized frequency 0.6) may also become unstable
since positive growth rate. If this is the case or not depends on the damping in the system.
• It is possible to find combinations of fuel time lags for main 1 respective main 2 that gives a
lesser growth rate.
• Having same fuel time lag for the two fuel feed lines gives among the highest growth rates while
a larger difference between the two feed lines seems to be beneficial for stability.
• The equivalence ratio fluctuations is dominated by the fluctuations of the air supply due to the
high value of the fuel line impedance.
4.2.7 Utilizing the Full Flame Model
The influence on stability due to both volume flow fluctuations as well as equivalence ratio fluctuations
has been studied separately in the preceding sections. This was found to be a powerful way to better
understand the different phenomena. Anyway, a study utilizing the full flame model at once was
performed using the best estimates available for the different time lags. This is a burner time lag
estimated from the bulk velocity and distance between the split plate and assumed position of the
flame. The fuel time lag characterizing equivalence ratio fluctuations for the two fuel feed lines were
estimated by particle tracing and CFD. The resulting stability plot is shown in figure 49 for different
load points. For the part load cases the fuel flow and total air mass flow from the compressor were
adjusted according to engine performance data. At lower loads, the mass flow will be lower while the
58
density goes up. To adjust the model for part load the time lags were thereby adjusted according to
the obtained change in velocity in the premixing channels. It is clearly noted that the growth rate
will be significantly lower for lower loads which is in line with operation experience. Moreover, at
part load the maximum Mach number in the guide vane will be lower and as shown in section 4.3.3
this will reduce reflections which will further decrease growth rate at part load. For this study the
reflection coefficient at the outlet has not been changed for the part load cases.
Figure 49: Stability plots. Left figure: Including both volume flow and equivalence ratio fluctuations.
Right figure: Including only equivalence ratio fluctuations
When evaluating the results it is of importance to consider the strength of the different terms/phe-
nomena in the flame model. In figure 49, stability plots when utilizing the full flame model and for
only including equivalence ratio fluctuations are shown. The stability plot only including volume flow
fluctuations is shown in figure 37. Comparing the results, the influence due to fluctuations of equiva-
lence ratio is for this configuration much lower than the influence due to the volume flow fluctuations.
If this is realistic or not was not possible to determine within this study. The relative strength will
also change depending on e.g. the fuel line impedance.
4.3 Measures to Improve Stability
In this section, a few measures to improve thermoacoustic stability are studied.
4.3.1 Change Combustor length
One way to improve stability is to change the acoustic properties of the system which can be done by
changing the length of the combustor cans. There are mainly two reasons stability may be improved
due to changed length. First, change of the mode shape may result in the heat release to be located in
a position with relative lower pressure amplitude for the particular mode. This will make less acoustic
energy to be transferred to the acoustic field according to Rayleigh’s integral. Second, changing the
length will change the eigenfrequencies and hence the convective times to the period of oscillation will
be different. The risk on the other hand is that modes that was stable instead will become unstable.
59
A study were performed in which the length of the can was increased respective decreased by 100mm.
The stability plot is shown in figure 50. Very small differences were seen for the first mode due to
the change of length being small compared to the wavelength. However, according to this model, the
length should not be increased since a slight increase in growth rate are predicted with increased can
length. This can be further understood from the mode shape of the first mode shown in figure 51. An
increased length will make the relative pressure within the mode to be higher at the location of the
flame.
Figure 50: Stability plot. Adjusted combustor length.
Figure 51: Mode shape for the first mode. Adjusted combustor length.
4.3.2 Including Helmholtz Resonators
Helmholtz resonators are common devices used in combustion chambers due to their high damping
effect close to the Helmholtz frequency (fH). Introduction of a Helmholtz resonator can damp a
combustion instability created elsewhere in the combustor. However, a Helmholtz resonator has a
60
very small operation window around fH and these devices must therefore be tuned for each frequency
of interest. An illustration of a Helmholtz resonator is shown in figure 52. For low frequencies when
the wavelength is much longer than the size of the resonator, wave propagation can be neglected and
the resonator will be equivalent to a mechanical mass spring system. The air in the neck will act as
the mass while the air inside the resonator will act as the spring, [23].
Figure 52: Schematic illustration of a Helmholtz resonator.
The Helmholtz frequency is determined from the geometric dimensions of the resonator as
fH =c
2π
√AnV0Ln
, (62)
where An is the neck cross-section area, Ln the length of the neck and V0 the volume of the chamber.
In the network code, a Helmholtz resonator can be modelled by duct and area change elements. A
small network model that only contain a resonator as shown in figure 52 was created and used for
tuning of the resonator dimensions. The temperature in the resonator was assumed to be the com-
pressor discharge air temperature at full load conditions. A circular cavity with both diameter and
length being 25cm was selected. The forced response option was used and the Helmholtz frequency
was determined from the impedance at the inlet to the resonator. At the Helmholtz frequency the
impedance will change phase (180◦). The Helmholtz frequency predicted by the code was found to
correspond very well to the analytical expression given in equation 62.
A Helmholtz resonator was now introduced at two different locations in the full model as shown
in figure 53. These locations were both outside the can since introduction of a Helmholtz resonator
of this size directly in the combustion can is not feasible. Resulting stability plot is shown in figure
54 and the mode shape of the first mode is shown in figure 55.
61
Figure 53: Schematic illustration of the network model, Helmholtz resonator positions.
Figure 54: Stability plot with a Helmholtz resonator in two different locations.
It was found that the introduction of a Helmholtz resonator in position 1 has very small influence on
stability. The main reason for this is that the pressure amplitude in that position is low for the mode
of interest. Introducing the Helmholtz resonator in position 2 on the other hand gives a significantly
lower growth rate for the first mode.
62
Figure 55: Mode shape for the first mode with a Helmholtz resonator in two different locations.
4.3.3 C-stage
Stability as well as emissions can be improved by introduction of a fraction of the main fuel upstream
the burner. Within Siemens this is commonly referred to as c-stage and is further described by Siemens
AG, [12]. An approximate location of c-stage fuel injection is illustrated in figure 47. Fuel injection
upstream the burner allows for a better mixing of the fuel and oxidizer which is good in order to
lower emissions. On the other hand, the amount of fuel introduced must be low enough to not give a
combustible mixture for safety reasons.
From a stability perspective, one more fuel injection location open for a further distribution of the
convective time lags. Furthermore, inside the swirler where the main fuel is normally injected the flow
velocity is significantly higher than in the casing upstream the burner. This means the fuel injection
holes must be moved a longer distance inside the swirler than if located in the casing to accomplish
the same change in convective time lag. In general the swirlers are not easily changed due to they
have been carefully optimized for mixing performance. Additionally, for low frequencies featuring
relatively long oscillation periods, the change of time lag required to see any significant difference can
not be accomplished in the swirlers considering the size and velocities. In the casing on the other
hand, the velocities are much lower and significant convective time lag changes can be achieved by
a much smaller change of injection location. To conclude, the c-stage allows for further distribution
and careful optimization of convective time lags.
The network model was adjusted and 10% of the main fuel was introduced as C-stage for the SGT-750
full load conditions. Resulting stability plot for different values of the c-stage fuel time lag (τi,c) is
shown in figure 56 together with the case without any C-stage fuel injection. Only equivalence ratio
fluctuations were included in this study.
63
Figure 56: Stability plot for 10% c-stage, different values of the c-stage fuel time lag.
To conclude, there are c-stage fuel time lags which gives a significantly lower growth rate compared
to no c-stage. However, the results also indicates a higher mode (around normalized frequency 0.6)
that may become unstable if the c-stage fuel time lag is not carefully selected.
4.4 Conclusions from the SGT-750 Study
• Evaluation of measurement data from a full engine test showed two distinct peaks close in
frequency dominating the thermoacoustic response at low frequencies.
• Further investigation of coherence and transfer function phase showed the first peak most prob-
ably is a global mode including all eight cans. The second peak was found to not be correlated
at all between the cans and hence a local phenomena in each can.
• A network model of one of the identical cans has been developed and validated against available
FEM calculations. Longitudinal modes are captured well.
• The assumption of the combustor outlet boundary being a hard wall has been shown to be
reasonable for low frequencies. Also, the full global model was found to not be very sensitive to
this boundary condition considering eigenfrequencies. Growth rate will be lower for lower values
of the outlet reflection coefficient since more acoustic energy leaves the system.
• The first instability (lower frequency) peak seen in the measurements was not found by the
network model of one single can which confirms the mode being a global mode.
• An unstable mode was predicted by the network model at the frequency of the second peak seen
in the measurement data. This indicates the second peak is most probably a longitudinal mode
in each can.
64
• Investigation of the mode shape showed the single can mode has a high pressure amplitude in
the location of the flame and are hence easily triggered by a heat release source. The mode was
triggered by all reasonable time lag vales tested.
• It has been shown that both Helmholtz resonators and c-stage may be used to improve stability
while reasonable changes of can length would have small influence at those low frequencies.
However, a more detailed study need to be performed in order to make a well justified design
change.
65
5 Discussion
In this work, the phenomena of thermoacoustic instabilities has been explored and the underlying
mechanisms have been discussed. A generalized network code developed by Siemens AG has been
used to study thermoacoustic stability of the SGT-750 gas turbine combustor in the low frequency
range. In addition, measurement data has been evaluated and the prediction by the network code was
found to be in good agreement with the measurements.
Many assumptions has been made through out this work which was required in order to get to
the final model. The very first assumption is linear theory. The underlying equations are linearized
expressions even though the presence of a heat release zone and large amplitude acoustic oscillations
violate the linear assumption. The linear theory is still useful to predict critical frequencies but the
limitation must be remembered. Negligible viscous damping and 1-dimensional acoustics was also
assumed and justified by the argument of low frequencies. The most severe assumption utilized is
probably about the flame. The total flame heat release was assumed to be concentrated to a discrete
axial location while the real flame shows a significant axial distribution. Regarding the time lags, they
were assumed to be completely dominated by convective times and the relevance of including other
smaller contributions to the time lags were shown to be small by sensitivity studies. The flame model
utilized could not be verified to actually be an acceptable representation of the real flame within the
frames of this thesis work. It should be mentioned that this flame model may not be appropriate at
all for the SGT-750 burner. Moreover, both volume flow fluctuations at the assumed reference plane
and fluctuations in equivalence ratio was found to trigger the mode. It could not be concluded which
of the phenomena or a combination that is the responsible mechanism involved.
Parameter studies were found to be very useful to investigate trends in the network model. How-
ever, one should carefully define what the purpose is before hand and not run too many cases since
that will be more confusing than helpful.
The coupled modelling approach utilized in the network code was found to have both advantages
and disadvantages. The main advantage is that e.g. velocities and speed of sound are calculated for
the actual temperature, pressure and species by the code which otherwise have to be calculated in
a separate model. On the other hand, it requires relevant flow features to be carefully considered
in the acoustic model. Adapt the model to capture the flow field was sometimes found to be very
time consuming and convergence errors in the flow solver had to be sorted out several times. The
code was found to have some issues with singularities and excluding functions such as pressure losses
or mass flows by setting values to zero should be avoided. Instead very small values should be used
to make the influences negligible. At present, there are a lack of people using the network code for
thermoacoustic studies, by more users, the functionality and flexibility could be further improved.
Verification of different sub-parts within the model as well as the assumptions has been difficult
66
through out the work. One example is the flame and another is the combustor outlet boundary con-
dition which was modelled without any real verification. The verification issue originates in acoustic
measurements for a real gas turbine combustor at operation conditions are not easily performed in
practice.
Linear network models like the one used here will not give precise and accurate numerical answers
but is useful to predict trends and gain understanding. This means that expectations on these mod-
els must be adapted accordingly. However, the network modelling approach in combinations with
FEM, CFD and measurements constitute powerful tools to understand and predict thermoacoustic
instabilities.
6 Recommended Future Work
• Most important for further investigation is characterization of the flame. The suggestion is to
study the flame transfer function by using reactive flow and LES simulations. Knowing the
characteristics of the flame would give an indication of which flame model is really suitable for
the SGT-750.
• Compare results with measurement data from a single burner/can rig test. This may confirm if
the first peak is a global mode including all cans as found in this study.
• Add the functionality in the generalized network code to have more than one flame region. With
the possibility of several flames, the full combustor featuring all eight cans could be modelled
and global modes could be studied.
• Apply the network approach to other Finspong engines featuring annular combustors. This is
currently not supported in the generalized network code used here. However, there are other
network codes within Siemens that can handle annular combustors which should be explored.
67
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69
APPENDIX
A The Two Microphone Method
To determine the amplitude of the right respective left travelling wave in a 1-D acoustic field with
only plane waves, the two microphone method can be used. The acoustic field when a mean flow is
present and with the time dependence eiωt can be described as
p(x) = p+e−ik+x + p−e
ik−x. (A-1)
To determine the two unknowns p+ and p−, two microphones located a distance L apart can be used,
see figure A-1.
Figure A-1: Schematic illustartion of the two microphone method.
The acoustic pressure each of the microphones will be exposed to can from equation A-1 and figure
A-1 be written as
p1 = p+ + p−
p2 = p+e−ik+L + p−e
ik−L.(A-2)
Solving for p+ and p− gives
p+ = D−1(p1eik−L − p2)
p− = D−1(−p1e−ik+L + p2),
(A-3)
where D is given by
D = eik−L − e−ik+L = 2i · exp(iMkL
1−M2
)sin
(kL
1−M2
). (A-4)
There is a singularity when kL/(1 −M2) = nπ for n = 0, 1, 2... This is equivalent to the distance
between the microphones equals a multiple of half the wavelength. For practical reasons the method
should therefore be used in the frequency range
0.1π ≤ kL
1−M2≤ 0.8π. (A-5)
Recommended