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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[3] Abom M., ”An introduction to Flow Acoustics”, 4th edition, The Marcus Wallenberg Laboratory

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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)