Investigation of Combustion Instability in a Single Annular Combustor

University of Cincinnati

Date: 10/29/2010

I, Fumitaka Ichihashi , hereby submit this original work as part of the requirements for the degree of Master of Science in Aerospace Engineering.

It is entitled: Investigation of Combustion Instability in a Single Annular Combustor

Student's name: Fumitaka Ichihashi

This work and its defense approved by:

Committee chair: San-Mou Jeng, PhD

Committee member: Shanwu Wang, PhD

Committee member: Kelly Cohen, PhD

Committee member: Asif Syed, PhD

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Last Printed:3/4/2011 Document Of Defense Form

Investigation of Combustion Instability in a Single Annular Combustor

A thesis submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of

Master of Science

in the Department of Aerospace Engineering and Engineering Mechanics of the College of Engineering Nov 2010

by Fumitaka Ichihashi

B.S., Aerospace Engineering, University of Cincinnati, USA

Committee Chair: Dr. San-Mou Jeng

Abstract

The well known criterion for combustion instability is called the Rayleigh‘s criterion. It indicates that, for combustion instability to occur, the heat release rate (q‘) and pressure oscillation (p‘) must be in phase.

This thesis describes measurement techniques and study methods for combustion instabilities that occurred in the prototype single annular sector Rich-Burn Quick-Mix Lean-Burn (RQL) combustor on the original (short) and new (long) experimental rig configuration with a focus on q‘ and p‘ measurements. A change in the configuration of the combustor rig was necessary in order to acquire more precise measurements of forward- and backward-moving acoustic pressure waves within the rig by mounting pressure transducers on preselected locations of the upstream duct, downstream duct and combustion area. Pressure transducers provided such local pressure behaviors as amplitude and frequency per location, also in addition to transfer functions that allow for the calculation of the acoustic impedance at any location within the combustor rig. A high-speed camera was capable of filming a chemiluminescene image, i.e., the rate of heat release through a quartz window that is mounted on the side of the combustor.

Two imaging analysis techniques, Proper Orthogonal Decomposition and Fourier Transformation, were applied to the chemiluminescene image obtained by a high-speed video device. Two different test cases were investigated. Both a high and low fuel-to-air ratio were used for the investigation of the Rayleigh‘s criterion, which was confirmed by the corresponding q‘ and p‘ data sets. Finally, the resonance frequency that agrees with combustion instability was well predicted by utilizing the one-dimensional wave propagation theory and the known geometry of the combustor rig, temperature of fluid, and boundary conditions.

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Acknowledgements

I would like to first state my deepest appreciation to my advisor, Dr. San-Mou Jeng, for providing me funding, guidance, and the opportunity to work on the combustion instability rig, and also on other projects and rigs at the University of Cincinnati Center Hill combustion research laboratory. I have learned extremely valuable engineering knowledge through these experiences. I would like to thank Dr.

Kelly Cohen for guiding and supporting my thesis. His wisdom and passion have educated me in additional engineering fields of analysis and control systems; he has also coached me on my work ethic so that I could become a better engineer. My great appreciation for him will never be forgotten. Finally, I would like to express my great appreciation to Dr. Asif Syed for supporting my thesis and being a mentor to me for five years while at college. The experiences, knowledge, wisdom, and ethics that I have learned from him has become the core of my engineering work. My deepest appreciation also goes to Dr. Shanwu

Wang of GE Aviation, for his time, effort, and advice as a committee member. Additionally, I would like to thank Mr. Curtis Fox for supporting and educating me on engineering measurements and programming throughout college life. Without him, I would not have enjoyed research and obtained the confidence I have now. My thanks also go to Dr. Samir Tambe and University of Cincinnati Aerospace Engineering undergraduate student, Joe Tscherne, for helping me set up and run the combustion instability experimental rig, and I am very thankful to University of Cincinnati Aerospace Engineering undergraduate student, Christine Englert, for enhancing my thesis writing.

Table of Contents Abstract ...... i Acknowledgements ...... iv Table of Contents ...... v List of Figures ...... vii Nomenclature ...... xii Chapter 1: Introduction, Literature Review, and Objectives: ...... 1 1.1 Introduction: ...... 1 1.2 Literature Review: ...... 8 1.3 Objectives: ...... 14 Chapter 2: Experimental Setup ...... 15 2.1 Prototype Single Annular Combustor Sector ...... 15 2.2 Research Facility ...... 15 2.2.1 Pneumatic System ...... 16 2.2.2 Fuel System and Ignition ...... 16 2.3 Experimental Section ...... 17 2.4 Instrumentation System ...... 18 Chapter 3: Chemiluminescene – POD and Video FFT (Combustion) ...... 33 3.1 Theoretical Notes ...... 34 3.2 FFT Analysis:...... 36 3.3 POD Analysis: ...... 40 Chapter 4: Acoustic Wave (FFT and Impedance)...... 64 4.1 TEST Results and Discussion – Original Rig (Short) Configuration: FFT ...... 64 4.2 Conclusion of Test Results from the Original Short Rig Configuration: ...... 70 4.3 Suggested Changes in the Original (Short) Rig Configuration: ...... 71 4.4 Results and Discussion – New Rig (Long) Configuration: FFT ...... 71 4.5 Pressure Dynamics Measurement ...... 76 4.5.1 Acoustic Transmission and Impedance: Step Area Change and Gradual Area Change ...... 76 4.5.2 Upstream Swirler Impedance Measurement (Non-Reacting Case): ...... 82 4.5.2.1 Validation of Upstream Transducers: ...... 82 4.5.2.2 Impedance Measurement: Upstream Duct ...... 85 4.5.2.3 Impedance Measurement: Step Change at Flange: ...... 87 4.5.2.4 Impedance Measurement: Diffuser and Swirler ...... 89 4.5.3 Downstream Swirler Impedance Measurement (Non-Reacting Case): ...... 91 4.5.3.1 Validation of Downstream Transducers: ...... 91 4.5.3.2 Impedance Measurement: Downstream Duct ...... 93

4.5.3.3 Impedance Measurement: Combustor...... 95 4.5.4 Swirler Impedance ...... 99 4.5.4.1 Swirler Impedance by Acoustic Pressure Wave ...... 99 4.5.4.2 Swirler Impedance by Pressure Drop ...... 101 4.5.5 Pressure Wave Measurement in the Combustion Area ...... 103 4.5.5.1 Plane Wave Measurement Using Top Half of PCB Transducers on the Combustor Wall105 4.5.5.2 Plane Wave Measurement Using Bottom Half of PCB Transducers on the Combustor Wall ...... 108 4.5.5.3 Plane Waves and First Transverse Mode Measurement in the Combustor ...... 111 4.5.6 Upstream and Downstream Boundary Condition: ...... 114 4.5.6.1 Air Heater Impedance Calculation: ...... 115 4.5.6.2 Upstream Mass Flow Control Valve ...... 119

4.5.6.3 Exhaust Duct (X5) Radiation Impedance Calculation ...... 124 4.5.7 Impedance Calculation with Moving Fluid...... 127 4.5.7.1 Effect of Fluid Flow within Experimental Range ...... 127 4.5.7.2 Effect of Air Temperature on Impedance: ...... 131 Chapter 5: Result and Discussion – Rayleigh‘s Criterion (p‘ and q‘) ...... 140 Chapter 6: Modeling of Acoustic Resonance Modes ...... 143 6.1 Introduction: ...... 143 6.2 Modeling of Acoustic Resonance Criterion / Theory: ...... 144 6.3 Acoustic Resonance Modeling Result: ...... 147 Chapter 7: Suggested Future Work ...... 150 7.1: Fuel, Swirler, and Test Conditions ...... 150 7.2: Changing Impedance of Pressure Wave Path ...... 150 7.3: Feedback Loop ...... 151 Conclusion ...... 154 Reference ...... 156 Appendix I ...... 161 Appendix II ...... 167 Appendix III ...... 172

List of Figures

Figure 1.1 Acoustic Emissions of the Combustor Sector, CH4 and C3H8 …3 Figure 2.1 Combustor Sector Hardware …15 Figure 2.2 Schematic of Experimental Rig and Facility Set Up …20 Figure 2.3 Original (Short) Combustion Rig Configuration …21 Figure 2.4 New (Long) Combustion Rig Configuration …22 Figure 2.5 Wiring Diagram 1 …23 Figure 2.6 Wiring Diagram 2 …24 Figure 2.7 Heater Core Picture …25 Figure 2.8 Schematic of Heater …25 Figure 2.9 Cooling Air System for a Speaker …26 Figure 2.10 Upstream Section of New (Long) Combustion Instability Research Rig …26 Figure 2.11 Combustor Section of New (Long) Combustion Instability Research Rig …27 Figure 2.12 Downstream Section of New (Long) Combustion Instability Research Rig …27 Figure 2.13 Cooling Air System for a Speaker …28 Figure 2.14 Micro Motion CMF050 Coriolis Flow Meter …28 Figure 2.15 Homemade Ignition System …29 Figure 2.16 Data Acquisition System …29 Figure 2.17a Three Dimensional Array of Frames to Frequency …30 Figure 2.17b Three Dimensional Array of Frames to Frequency …30 Figure 2.18 Schematics of Pressure Sensor on Acoustic Measurements …31 Figure 2.19 Amplitude and Phase Response of Two Pressure Sensors …32 Figure 3.1 A Frame of High Speed Video …46 Figure 3.2 Acoustic Emission Spectrum …46 Figure 3.3 Mean of Energy Release …46 Figure 3.4 r.m.s. of Energy Release …46 Figure 3.5 Energy Release Spectrum …46 Figure 3.6 Amplitude Contours of Energy Release at 277.3, 279.5, and 281.7 Hz …47 Figure 3.7 Phases Angle (Radian) Contours at 279.5 Hz …47 Figure 3.8 One Cycle of Energy Release Motion at 279.5 Hz (12 Plots with 30° Increments) …48 Figure 3.9 (Approximate) One 279.5 Hz Cycle of Original Video on Fluctuating Energy Release Component (Start with Frame 200 with Frame Increment of 2 or Phase Angle Increment of 30.5 Degree) …49

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Figure 3.10 POD Mode #1 Contours and Associated Temporal Spectrum …50 Figure 3.11 POD Mode #2 Contours and Associated Temporal Spectrum …50 Figure 3.12 POD Mode #3 Contours and Associated Temporal Spectrum …50 Figure 3.13 POD Temporal Coefficients as a Function of Time …51 Figure 3.14 POD Accumulated Mode Energy …51 Figure 3.15 (Approximate) One 279.5 Hz Cycle of POD Mode #1 Reconstruction (Start with Frame 200 with Frame Increment of 2 or Phase Angle Increment of 30.5 Degree) …52 Figure 3.16 (Approximate) One 279.5 Hz Cycle of POD Mode #2 Reconstruction (Start with Frame 200 with Frame Increment of 2 or Phase Angle Increment of 30.5 Degree) …53 Figure 3.17 (Approximate) One 279.5 Hz Cycle of POD Mode #2 Reconstruction (Start with Frame 200 with Frame Increment of 2 or Phase Angle Increment of 30.5 degree) …54 Figure 3.18 (Approximate) One 279.5 Hz Cycle of Sum of First Five POD Reconstruction (Start with Frame 200 with Frame Increment of 2 or Phase Angle Increment of 30.5 degree) …55 Figure 3.19 Acoustic Emission Spectrum from Case 2 and Case 3 …56 Figure 3.20 Amplitude and Phase Contours of Energy Release Rate at 106, 208, and 354Hz, Case 2 …57 Figure 3.21 POD Mode #1, #2, and #3 Contours and Associated Temporal Spectrum, Case 2 …58 Figure 3.22 POD Temporal Coefficients as a Function of Time, Case 2 …59 Figure 3.23 POD Accumulated Mode Energy, Case 2 …59 Figure 3.24 Amplitude and Phase Contours of Energy Release Rate at 106, 214, 322, and 428Hz, Case 3 …61 Figure 3.25 POD Mode #1, #2, and #3 Contours and Associated Temporal Spectrum, Case 3 …62 Figure 3.26 POD Temporal Coefficients as a Function of Time, Case 3 …63 Figure 3.27 POD Accumulated Mode Energy, Case 3 …63 Figure 4.1 Power Spectrums of Microphone and Four PCB Pressure Transducers …64 Figure 4.2 Energy Release Rate at 273.4 Hz, Case 1a …66 Figure 4.3 Energy Release Rate at 442.8 Hz, Case 1a …67 Figure 4.4 Power Spectrum of Microphone and Four Pressure Sensors …68 Figure 4.5 Energy Release Rate at 282.8 Hz, Case 1b …69 Figure 4.6 PCB Transducers on the Combustor Wall …72 Figure 4.7 Power Spectra of the Microphone and the PCB Transducers, Case 2 …73 Figure 4.8 Power Spectrum of Microphone and PCB Transducers, Zoomed In, Case 3 …74 Figure 4.9 Power Spectrum of Microphone and PCB transducers, Full scale, Case 3 …75 Figure 4.10 Schematic of Upstream Duct and Combustor …76

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Figure 4.11 Step Area Change …77 Figure 4.12 Gradual Area Change …78 Figure 4.13 Upstream Calibration Check Schematic …82 Figure 4.14 Comparison of the Theoretical and Measured Reactance at x=1.016m (40‖) …83 Figure 4.15 Comparison of the Theoretical and Measured Transfer Functions Between PCB #2 and #1 …84 Figure 4.16 Comparison of the Theoretical and Measured Transfer Functions Between PCB #3 and #1 …84 Figure 4.17 Upstream Schematic (Non-Reaction Case) …85

Figure 4.18a Impedance of X1U before Step Change …87

Figure 4.18b Impedance of X1D after Step Change …88 Figure 4.19 Schematic of Diffuser Section …89

Figure 4.20 Impedance of the Swirler, ZSU, Measured from Upstream …90 Figure 4.21 Schematic of Downstream Duct and Combustor …91 Figure 4.22 The Comparison of Theoretical and Measured Acoustic Reactance Result Verifies The Correct Function of the PCB Transducers #4, #5, and #6. …92

Figure 4.23 The Acoustic Impedance, Z4, at the Inlet, X4, of the Downstream Duct …94 Figure 4.24 Schematic of Combustor Section …95 Figure 4.25 Schematic of Combustor Cooling Liner Section …96

Figure 4.26 Impedance of the Swirler, ZSD, Measured from Downstream …98 Figure 4.27 Impedance of the Swirler, ΔZ, Full Frequency Scale …99 Figure 4.28 Impedance of the Swirler, ΔZ, Selected Frequency Range …100 Figure 4.29 Pressure Drop Across Swirler vs. Mass Flow Rate …101 Figure 4.30 Resistance of the Swirler Obtained from Pressure Drop …102 Figure 4.31 Schematic of Downstream Duct and Combustor with Four PCB Transducers …103 Figure 4.32 Schematic of PCB Transducers on the Combustor Wall …104 Figure 4.33 Top Half of PCB Transducer on the Combustor Wall …105

Figure 4.34 Theory vs. Measurement of Reactance Measured at X1C Using Top Half Section …107 Figure 4.35 Bottom Half of PCB Transducer on the Combustor Wall …108

Figure 4.36 Theory vs. Measurement of Reactance Measured at X1C Using Bottom Half Section 110 Figure 4.37 Schematic of Combustion and Propagating Waves …111

Figure 4.38 Theory vs. Measurement of Reactance Measured at X1C Using all Four Transducers 112 Figure 4.39 Measurement of Upstream Boundary Condition, Air Heater …114

Figure 4.40 Measured Impedance at X0D, the Upstream Boundary Condition …118

Figure 4.41 Schematic: Mass Flow Control Valve Location …119 Figure 4.42 Resistance of Heater vs. Mass Flow Control Valve Openings …119 Figure 4.43 Reactance of Heater vs. Mass Flow Control Valve Openings …120 Figure 4.44 Schematic of Heater Section …121 Figure 4.45 A Hard Wall s Placed at Upstream End Section of the Air Heater …122

Figure 4.46 Reactance of the Section X-1 …122 Figure 4.47 Exhaust Duct End …124 Figure 4.48 Measured and Theoretical (Unflanged Circular Duct) Radiating Impedance …125 Figure 4.49 Measured and Theoretical (Unflanged Circular Duct) Radiating Impedance …126 Figure 4.50 Schematic of Entire Combustor Rig …127

Figure 4.51 Resistance at X4 vs. Various Mass Flow Rates …128

Figure 4.52 Reactance at X4 vs. Various Mass Flow Rates …129

Figure 4.53 Resistance at X5 vs. Various Mass Flow Rates …129

Figure 4.54 Reactance at X5 vs. Various Mass Flow Rates …130 Figure 4.55 Schematic of the Combustor Rig with Thermocouples Installed …131 Figure 4.56 Resistance of Heater vs. Various Air Temperatures …133 Figure 4.57 Reactance of Heater vs. Various Air Temperatures …133

Figure 4.58 Resistance at X1U vs. Various Air Temperatures …134

Figure 4.59 Reactance at X1U vs. Various Air Temperatures …134

Figure 4.60 Resistance at XSU vs. Various Air Temperatures …135

Figure 4.61 Reactance at XSU vs. Various Air Temperatures …135

Figure 4.62 Resistance at XSD vs. Various Air Temperatures …136

Figure 4.63 Reactance at XSD vs. Various Air Temperatures …136

Figure 4.64 Resistance at X4 vs. Various Air Temperatures …137

Figure 4.65 Reactance at X4 vs. Various Air Temperatures …137

Figure 4.66 Resistance at X5 vs. Various Air Temperatures …138

Figure 4.67 Reactance at X5 vs. Various Air Temperatures …138

Figure 4.68 Resistance as a Function of Wave Number at XSD Vs. Various Air Temperatures …139

Figure 4.69 Reactance as a Function of Wave Number at XSD Vs. Various Air Temperatures …139 Figure 5.1 p‘ and q‘ Phase Difference and Normalized Peak Frequency Amplitude for Case 2 …142 Figure 5.2 p‘ and q‘ Phase Difference and Normalized Peak Frequency Amplitude for Case 3 …142 Figure 6.1 Schematic of the Rig within Boundaries and Acoustic Source Location …144

Figure 6.2 Illustration of Discrete Combustion Location …145 Figure 6.3 Temperature Segments in the Combustor Rig …147 Figure 6.4 Instability Prediction from Model Using Measured Boundary Conditions …149 Figure 6.5 Instability Prediction from Model Using Simplified Theoretical Boundary Conditions … 149 Figure 7.1 Schematic of the Piston Mechanism to Change Impedance of Pressure Wave Path …151 Figure 7.2 Feedback Diagram …153

Nomenclature

an = waveform, time dependent coefficient; n = 1, 2, 3, … M

A(n) = eigenvectors bn = complex number corresponding to amplitude and phase

C = correlation matrix f = the acoustic frequency i = 1

M = total frame number in high speed video

Z = high speed video data

λn = eigenvalues v = video file image

ω = the angular frequency

Øn = POD mode shape; n = 1, 2, 3 … M f/a = fuel to air ratio xn = location, distance from the reference line x0 p = acoustic pressure u = acoustic particle velocity

Z = acoustic impedance z = specific acoustic impedance

A = incident complex acoustic amplitude (forward going propagating plane wave amplitude)

B = reflected complex acoustic amplitude (backward going propagating plane wave amplitude)

k = wave number,

κ± = wave number for moving fluid.

ρ = density of air

xii c = speed of sound

R = reflection coefficient

Hm,n = transfer function of transducer ‗m‘ with reference to ‗n‘

ARm,n = cross sectional area ratio,

Ln = distance between two locations (∆x)

Res = resistance m = mass reactance r = radius

G(x) = Rayleigh index

Tp = oscillation period q‘ = heat release oscillation p‘ = pressure wave oscillation

TF = transfer function

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Chapter 1: Introduction, Literature Review, and Objectives

1.1 Introduction

The phenomena of combustion ―Screech, Howl and Growl,‖ experienced in the combustors of gas turbine engines, are high amplitude pressure oscillations due to instability in the combustion process. These phenomena are also referred to as ―combustion dynamics.‖

The coupling of an unsteady heat release in the flame with the natural resonance of an acoustic mode in the combustor duct is believed to be the cause of this instability. The frequency and the high amplitude of the combustion screech phenomenon can cause severe structural vibrations and may result in the structural failure of some engine components. Thus, uncontrolled combustion dynamics can greatly limit the operability and performance of an engine. While the demand for the low emission combustion that is achieved by lean burn/low temperature combustion is increasing, one major tradeoff of lean burn combustion is that combustion instabilities, and the mechanisms causing combustion instabilities in gas turbine combustors, are not well understood. Although this topic has been widely studied by scientists and engineers, Hersh, et al [1], Yi, et al [2-4], Mohammad, et al [5], Cai, et al [6], Lieuwen [7-8], Johnson [9],

Zinn [10], Bellow [11], and Culick [12-13], the added geometric complexities and injector configurations of a practical combustor make their dynamical behavior currently unpredictable. Current design techniques in the industry are largely empirical and not clearly defined with respect to combustion instabilities.

This thesis describes investigation and measurement techniques of combustion instability in a ―Rich-

Burn, Quick Mix, Lean-Burn‖ (RQL) gas turbine engine combustor for a selected fuel mixture and inlet air temperature. In the following sections of this thesis describe the test rig and the measurement techniques used in the testing to define the acoustic field at all locations of interest in the test rig. The

1 thesis presents acoustic data from experiments conducted on this apparatus and discusses the acoustic pressure data measured in the combustor.

Acoustic Waves:

In his report, ―Issues in Combustion Noise,‖ Mani [14] asserted: ―the inclusion of ‗boundary conditions‘ imposed by turbo-machinery is an essential ingredient of a rational approach to the physics and calculation of combustion noise.‖ This is also true in regard to the understanding and analysis of the combustion-driven resonance phenomena observed in the experimental test rig used by the author at the

University of Cincinnati (UC). Therefore, the boundary conditions of the experimental rig were carefully determined by using arrays of pressure transducers. The acoustic impedance at the upstream termination

(just downstream of the swirler) clearly depends on the geometry of the upstream ducting that contains the preheated airflow. The acoustic impedance at the downstream end depends on the geometry of the exhaust opening, the geometry of the combustor, and the aero-thermo operating conditions in the combustor. The combination of the array of pressure transducers placed on the upstream and downstream of the combustor is capable of defining the impedance of the swirler. The impedance characteristic of the swirler is not only necessary while modeling the resonance phenomenon of the combustion rig, but it also shows acoustical attenuation in the frequency domain that gives researchers a better understanding and insight into the combustion research rig. Pressure transducers mounted on the combustor wall provide critical key information in identifying and proving Rayleigh‘s criterion.

Figure 1.1 Acoustic Emissions of the Combustor Sector, CH4 and C3H8

Acoustic Emission of Combustor:

The acoustic emissions of this combustor sector were surveyed by a microphone for a wide range of equivalence ratios, air inlet temperatures, and air pressure drops. The acoustic power spectral data at the different overall equivalence ratios are illustrated in Figure 1.1. Note that the results shown in Figure 1.1 are observed from the original (short) combustion rig configuration. Two high amplitude low-frequency modes (a higher overall equivalence ratio condition with frequency around 400Hz, and a lower overall equivalence ratio condition with an instability frequency around 200 Hz) were measured. Lower amplitude acoustic emission was also measured around 650 Hz, which corresponded to the ¼ acoustic wavelength based on the distance from the dome plane to the combustor exit plan and the temperature in the combustor.

The two potential instabilities (at 200 Hz and 400 Hz) were associated with different flame structures.

The typical amount of air in the combustor dome (dome cooling air and swirling air) is about 30% of the total airflow rate. For lower overall equivalence ratio conditions (<0.33), the acoustic mode is solely generated by heat release oscillations in the dome region, since the equivalence ratio within the dome is fuel-lean (fuel-lean mode). For higher overall equivalence ratio conditions (>0.33), the acoustic model is

3 associated not only with unsteady dome combustion but also with the complex heat release process associated with the dilution air jets, and partially-burned fuel-rich and partial mixture from the dome flows (fuel rich mode). A two-dimensional distribution of the unsteady heat release amplitude and phase angle for each unstable mode indicated that the fuel-lean mode and fuel-rich mode were driven by two totally different mechanisms. The transition from fuel-lean to fuel-rich instability featured a shift of driving mechanism. This study shows that the coupling mechanisms leading to low-frequency combustion instabilities are not unique and illustrates the difficulty of devising predictive models.

Acoustic Resonance Modes:

From the dimensions of the combustor test rig, the ducting of the rig could only allow the plane mode of acoustic wave propagation throughout the test rig over the frequency range of interest (<1000 Hz.).

Therefore, the development of the modeling for resonance phenomena based on the one-dimensional wave propagation theory became possible. It includes complexities of gas flow, variable area of duct cross-section, and the acoustic impedance characteristics at specified locations in the rig. The details of this theoretical model are described in the Appendix II or [28].

Chemiluminescene Imaging Technique:

Chemical reactions of combustion create entropy waves, which cause the generation of acoustic pressure waves. The high-speed digital charge-coupled device (CCD) camera video data show the location of the source of entropy oscillations at the frequencies of the measured high amplitude acoustic pressure oscillations. The acoustic power spectrum may contain a single frequency peak, or peaks at several frequencies. The resonance frequency of the combustor depends on the test rig structural configuration, as well as on the type of fuel, the flame location, and the boundary conditions at the upstream and downstream terminations. The relation of entropy oscillations and the acoustic excitation frequencies in the combustion chamber were studied by Ichihashi, et al [15] with a high-speed CCD camera.

To improve our understanding of the stability properties in such complex systems, encountered in many industrial applications, the flame structure of an atmospheric swirl-stabilized burner, incorporating dilution and cooling air holes and fed with natural gas fuel, was systematically investigated for various inlet temperatures, pressure drops, and air-fuel ratios. Stable and unstable regimes, which depend on the inlet air temperature, velocity, and the equivalence ratio, have been identified. Chemiluminescence imaging with a high-speed CCD camera, and simultaneous dynamic pressure measurements inside the combustor and in the upstream ducting, were used to characterize the combustor behavior. Imaging of the combustion field provided an insight on the flame structure and its interaction with the entering dilution air jets. For stable combustion, the flow field was characterized by the presence of intense heat release located on both the primary zone and the dilution jet impingement region.

Proper Orthogonal Decomposition:

This thesis describes the procedure of two different methodologies: temporal Fast Fourier Transform

(FFT), and spatial Proper Orthogonal Decomposition (POD), to analyze high speed videos on energy release dynamics (chemiluminescene). The case study was conducted on fuel-lean combustion dynamics with a single dominant acoustical frequency around 280 Hz. The temporal and spatial characteristics of the energy release dynamic revealed by both methods are illustrated and compared.

A common method used to substantially reduce the order of the model is proper orthogonal decomposition (POD). POD techniques are an efficient means of reducing spatially highly complex flow fields, by representing them as a small number of spatial modes and their temporal coefficients. This method is an optimal approach, because it will capture the largest amount of the flow energy in the fewest modes of any decomposition of the flow (Berkooz et al. [16]). POD computes a small number of orthogonal basis functions that contain as much information as needed to represent the original system dynamics. This is accomplished by selecting a set of bases, or modes, that contain the ―most energy‖ for a particular flow regime. By projecting the data onto the modes, the ensemble can be approximately

5 reconstructed, and thus POD enables analysis of all the necessary data in as few modes as possible. POD acts as a statistical tool for the construction of the complex flow field, from the Navier-Stokes equations, of low-dimensional dynamical models for the interaction of essential structures. A common approach, referred to as the method of ―snapshots‖ introduced by Sirovich [17], is employed to generate the basic functions of the POD spatial modes from flow-field information, obtained using either experiments or numerical simulations. This approach to the modeling of the global wake behavior behind a circular cylinder was effectively employed by Gillies [18]. Details of POD and its application in fluid dynamics and turbulence can be found in the paper by Berkooz et al. [16] and Cohen et al. [19 -22]. In a recent paper, Siegel et al [23] developed an extension to the POD approach, referred to as ―double proper orthogonal decomposition‖ (DPOD), in which shift modes have been added to account for the changes in the flow due to transient forcing.

Blue and green color intensity levels of the high-speed CCD camera video data on natural gas fueled combustion represent the chemiluminesence of flame dynamics-energy release dynamics. Each pixel of video data in the time domain has its own waveform shapes. Identifying waveforms in each pixel of high speed video data lets us differentiate data by color intensity, so that a time domain waveform of color intensity is obtained. In other words, the frequency behavior and its corresponding region in the combustor can be defined by a two-dimensional image. Fast Fourier Transform (FFT) is an ideal method to determine the frequency behavior in a region of the image. The results obtained by the high speed camera correlated with the pressure transducer data. This is one of the methods to identify stability characteristics in such a complex system.

Proper Orthogonal Decomposition (POD) is a sophisticated method utilizing high speed video data that consists of a frame by frame, or snapshot, method, to analyze the measured color intensity (heat release rate) properties that are related to the acoustical behavior/combustion dynamics. POD makes it possible for us to create a space and amplitude contour for a specified frequency of interest by defining its mode

6 shape and the corresponding time-domain waveform. This can provide a detailed analysis of combustor characteristics that are directly related to acoustic phenomena/combustion dynamics. Because we can separate data spatially and by frequency, we can visually specify the characteristics of the energy release rate in the combustor.

1.2 Literature Review

Combustion Instabilities:

Lord Rayleigh is a pioneer of combustion instability. After his study of heat addition, which resulted in the production of acoustic waves, he enunciated his famous criterion (Lord Rayleigh [24] (1878, 1945, Vol. II, p.226)): If heat be communicated to, and abstracted from, a mass of air vibrating (for example) in a cylinder bounded by a piston, the effect produced will depend upon the phase of the vibration at which the transfer of heat takes place. If heat be given to the air at the moment of greatest condensation, or be taken from it at the moment of greatest rarefaction, the vibration is encouraged. On the other hand, if heat be given at the moment of greatest rarefaction, or abstracted at the moment of greatest condensation, the vibration is discouraged.

Engineers and scientists have accomplished various ways to investigate and make sense of the Rayleigh criterion. Tremendous numbers of studies have been invested into the combustion instability subject, and a number of their publications are available in such widely spread subject categories as experimental report, approach, and modeling.

Culick has investigated further into Rayleigh‘s criterion. His concern with direct transfer of heat to the mechanical energy of acoustical motion was described in his work [12, 30-32]. His focus of study was to fully reveal the formula of Rayleigh‘s criterion, and then show how it may agree with a combustion chamber pressure oscillation analysis. Difficulty in formulating Rayleigh‘s criterion is stated by Culick

[13]:

…the combustion instability phenomena are extremely complicated involving unsteady gas dynamics and the combustion of reacting flow system which themselves cannot be described theoretically in all necessary detail. Hence the development of analytical methods must be guided at all stages by observational results.

Mechanism of Combustion Instability:

Conditions causing combustion instability require that the fluctuating acoustic pressure and unsteady heat release must be in phase, known as Rayleigh‘s criterion, and the rate of energy addition must exceed energy dissipation [8]. However, Lieuwen [8] mentions that extensive studies are conducted by many researchers to define what is causing combustion instabilities, but it is a difficult task, because oscillations of the velocity, pressure, temperature, and reactants are simultaneously oscillating during combustion instability. As it was said, a number of characterizations involving operating conditions have been performed by researchers [33-39], as well as several theoretical studies [40-45], such as the understanding of the mechanism that drives instability and the modeling of prediction for their occurrence. However, there is no clear answer for what causes the onset these instabilities. Researchers [35, 36, 42-44] have concluded that instabilities are caused by heat release oscillations, but not enough data and evidence are available for these conclusions to be considered firm evidence.

Mongia, et al [46] studied and presented measurement results of fuel oscillation (mass fraction) frequency at the inlet of combustion that correlate with the combustion instability oscillation frequency.

Furthermore, this study performed by Lieuwen [8] after Mongia, et al confirmed that the reactive mixture composition oscillation relates to the combustion instability [46].

Lieuwen [8] demonstrated that combustion instability is likely caused by the oscillation of acoustic waves, heat release, and the equivalence ratio feedback system. Also, he showed that the mean flow velocity and swirler/fuel-injector location in the combustor are part of the instability parameters. He concluded from his study that equivalent ratio oscillation is likely to be the cause of combustion instability. He also strongly noted other researchers‘ conclusions that combustion instabilities are caused by other mechanisms, such as the periodic shedding of vortices.

Onset Condition for Combustion Instability:

Lieuwen [7] and Johnson [9] investigated the measurement of the stability margin; in other words, how close the combustion is getting to becoming unstable.

Johnson [9] demonstrated the stability margin of the combustor by employing an acoustic driver for driven flow and a fuel injector that is capable of pulsing in order to generate reaction-rate oscillations.

The stability margin is then calculated and determined from transfer functions between the response of the heat release rate (q‘) and the combustion flow (p‘), and vice-versa. His report concluded that the transfer function of p‘/q‘ is a function of the frequency only and independent of amplitude, but q‘/p‘ strongly depends on the amplitude. He suggested that if the transfer functions of q‘/p‘ and p‘/q‘ can be measured, then they could be used to define the amplitude and frequencies of an unstable combustor.

Lieuwen [7] studied the stability margin quantification, the objective of which is similar to Johnson‘s [9].

However, Johnson‘s [9] approach required external devices, such as an acoustic driver and fuel injector actuator, that are not suitable for a fielded system but for an experimental lab environment. Lieuwen‘s approach for determining the stability margin is to define the rate of acoustic damping, because when acoustic oscillations are damped, a stable combustor results. The common way to quantify the acoustical damping of the combustor/duct is to monitor the rate of decay by turning off the acoustic driving device, but this requires external disturbance, which is against the practicality that Lieuwen is pursuing.

Therefore, he has utilized the model of work by Zinn [47] and Culick [48], which is a description of a superposition of nonlinearly interacting oscillators in the form of acoustic oscillations in a combustion chamber. His result shows that the stability margin can be tracked by pressure oscillation information gained using the described model. As a conclusion to his study, Lieuwen emphasizes that an accurate mathematical model is required in order to achieve a quantitative determination of the damping coefficient. Yi [2] has continued the study, based on a low-order model and a correlation-function based procedure developed by Lieuwen [7]. In his approach, damping ratio computations are in the frequency

10 domain instead of in the time domain, which is Lieuwen‘s approach [7]. Results conclude that the results of Yi‘s approach are equivalent and consistent with the result of Lieuwen [7].

Bellows [11] studied the nonlinear interaction of forced oscillations and natural acoustic modes in an unstable combustor, which was originally reported by Lieuwen, et al [49]. Also, a phenomenon described by Nayfeh [50] is that of frequency locking, which is also known as the nonlinear oscillator phenomenon; it states that decreasing amplitude of the unstable mode is caused by increasing amplitude of the driven mode. Bellows‘ [11] results show an alteration of peak amplitude, bandwidth and/or frequency, due to introducing external force; in other words, the instability mode is reduced and shifts its frequency peak with the introduction of an external force, while quenching does not occur.

Wave Propagation and the Impedance Tube Technique:

Zinn [10] has applied the impedance tube technique to solid propellant driving when pressure and/or velocity disturbances are subjected to combustion. Solid propellant and lean, premixed prevaporized

(LPP) or RQL combustion are different, but essentially the technique can be applied to both for combustion instability observation. In Zinn‘s set up, the acoustic driver is used to create a standing wave, and an array of pressure transducers is mounted on the wall in order to measure wave propagation. This technique was originally established by acousticians [51, 52]. The impedance tube is used to measure acoustical attenuation of various materials by measuring incident and reflected waves. An array of pressure transducers mounted on the impedance tube is capable of instantaneous measurements that allow the observer to determine individual amplitudes and phases. Transfer functions obtained from the array of pressure transducer amplitudes, phase information, and geometrical information of pressure transducer locations, gives incident and reflected complex acoustic wave amplitudes. This technique was utilized for combustion instability by combustion researchers to understand an unstable rocket motor by Feiler, et al, and Temkin, et al [53, 54] in the first phase. Zinn [10] studied two conditions: combustion and no combustion. The results show that the impedance tube technique was capable of identifying that the

11 injectors acted as an acoustical damper when there is no combustion occurring, and that particular frequencies are amplified when there is combustion occurring.

Chemiluminescene Analysis Technique:

Yi [3] has investigated combustion using the chemiluminescene image analysis technique. His analysis in flame spectra within the UV/visible-light range from his research combustor using a spectrometer and intensified charge-coupled device (ICCD) camera gave insight into the heat release rate and chemiluminescene relationship. Yi has stated that a general assumption of proportionality in chemiluminescene and heat release rate is invalid, but proportionality could be valid if the weakly turbulent, or variation of equivalence ratio, and strong acoustic oscillations are absent.

Ichihashi [15] used a high-speed charge-coupled device (CCD) camera for chemiluminescene analysis.

He applied and illustrated two different mathematical approaches: Proper Orthogonal Decomposition

(POD) and Fast Fourier Transform (FFT), on the high-speed camera chemiluminescene image data to achieve insight into combustion instabilities. In his analysis, FFT gave the result of spatial distribution of amplitude and phase relation, while POD gave spatial mode shapes and energy, as well as temporal information of individual modes.

POD Technique:

As described in the introduction section, Proper Orthogonal Decomposition (POD) is a mathematical technique used to reduce highly complex data, such as a turbulent flow field, to small number of spatial modes and their time coefficients. Berkooz [16] stated that this is the optimal approach to describe, with the fewest modes of any decomposition of the flow for the largest amount of flow energy. In other words, complex information can be redrawn without losing its main or dominant characteristics by using the

POD technique. Sirovich [17] introduced a method called ―snapshots,‖ which generates the basis function

12 of the POD spatial modes from the flow field data. Ichihashi [15] used Sirovich‘s POD snapshot method to investigate combustion flow field data obtained by a high speed CCD camera.

Acoustics:

In order to understand combustion instability, a good understanding of pressure waves, i.e., acoustic waves, is required. Acoustical physics was well established years ago. In this section, the aspect of acoustic physics involved in this thesis is briefly discussed. Acoustic waves experience various geometrical changes from the upstream of the combustor to the exhaust. Lindsay [25] expressed the behavior and relation of acoustic waves and particle velocity during step area change and gradual area change in his published book. Also, Kinsler [26] showed the calculation of impedance, as well as the theory of radiating impedance, in his book: impedance of the exhaust section is not zero, but has radiating impedance that the researchers need to take into consideration.

Modeling of the Resonant Mode:

Syed [27] has proposed a model in which the one dimensional wave propagation theoretical method can be applied to show an acoustic liner, which has a step change of cross sectional area and resistance, to determine its impedance characteristic. The correlation between measurement and his model has shown excellent agreement. A one-dimensional model of the acoustic resonance modes in combustors is introduced in Ichihashi [28], and in Appendix II.

Simon [29] has shown a linear model for thermoacoustic oscillations in a lean, premixed prevaporized

(LPP) combustor to predict combustion instability in annular LPP combustors. His combustor is a straight duct with various diameters. His assumptions of constant cross-sectional area of duct and treating the inlet and outlet as open ends are the limitations in his model.

1.3 Objectives

Significant increase in acoustic emissions from the combustor at a specific fuel-air mixture condition was observed during the operation of the single annular sector combustor at the University of Cincinnati combustion research lab. As a first step, frequency peaks of acoustic emissions, using a microphone stationed near the combustor, were studied. The sharp peak in the frequency domain spectral plot from the microphone can be stated as an occurrence of combustion instability. The well known criterion of combustion instability, Rayleigh‘s criterion, is that the heat release and pressure wave are coupled, therefore the study then focuses on the observation of the flame structure correlating with acoustic emission and the pressure wave reacting with the heat release. It is important to note that the pressure wave depends on the rig structure; in other words, it depends on the boundary conditions of the upstream and downstream of the combustor rig.

The focus of this thesis is the establishment of a measurement technique, data acquisition programming, introduction of mathematical techniques for combustion instability analysis, and experimental rig construction at the University of Cincinnati combustion laboratory, while conducting a study of combustion instability behavior, such as identifying Rayleigh‘s criterion. The single annular sector combustor contains one plane with pressure transducers and another plane that is the quartz window for viewing. Since the viewing area is restricted to a one-side plane, heat release can be analyzed in a two- dimensional form. From the radiating sound wave measurement, the frequencies of interest are below

1000Hz. The geometry of the combustor cross section allows only plane wave propagation to occur below

1000Hz. Therefore, the one-dimensional acoustic wave propagation theoretical method can be applied to this study. However, complex higher order modes are believed to exist in the combustion area, so it is capable of measuring a higher order mode by considering the pressure transducer configuration.

Moreover, introduction to the acoustic resonance mode modeling procedure, closed feedback forced oscillation technique, and changes in boundary conditions will be discussed as future work.

Chapter 2: Experimental Setup

2.1 Prototype Single Annular Combustor Sector

The combustor used in the experiments described in this thesis is shown in Figure 2.1. It is a segment of a single-annular combustor from a prototype gas turbine engine. The photograph in Figure 2.1 shows the side of the combustor, with a stainless steel plate with four PCB pressure transducers installed in it, and the other side of combustor has the same size stainless plate, but this has a quartz window for viewing the combustion process.

Figure 2.1 Combustor Sector Hardware

2.2 Research Facility

The experiment required delivering air and fuel flow to the test combustor rig. In this section, the pneumatic system and the fuel system are considered as research facility standard equipment and described. A schematic of the air flow and fuel delivering systems are also illustrated with the experimental rig in Figure 2.2.

2.2.1 Pneumatic System

The manufacture and model name of the air compressor is Sullair LS10-40H. The air compressor is capable of delivering a flow rate of 156cfm and air pressure of up to 125psi. The compressed air from the compressor is filtered by a 20µm Van Air Systems filter element, then goes through a Pneumatech

AD150 40°F dew point dryer. The pressurized tank is rated for 200psi at 150°F. The pressure is regulated to approximately 80psi for the test cases described in this thesis. The air flow control valve regulates the flow rate while the rate is measured by a Micro Motion CMF050 Coriolis flow meter, as shown in Figure

2.14. The air flow control valve and the main experimental rig are connected by a two-inch flexible hose.

A Sylvania 36 kW (seen in Figure 2.9), a flanged inline heater, is responsible for heating the air to the desired temperature level. The pressure drop across the swirler cup is one of the parameters used to state the test criterion, and values in percentages are recorded by measuring the pressure difference between the upstream and atmosphere for the original (short) rig configuration, as in Figure 2.3, and for the new (long) rig configuration, shown in Figure 2.4, the upstream and the just exit of combustor using a Meriam differential pressure gauge, model 2110P-DI0200, that is rated 0-200 inch-H2O.

2.2.2 Fuel System and Ignition

The fuels chosen for the experiment introduced in this thesis are methane and propane gases, although using Jet-A aviation kerosene and natural gas are possible as fuel sources as is an application of water injection (a dual orifice swirler). The 100-pound tank containing liquid phase gas is provided for the experiment. First, gas will travel through a single stage regulator. Between the regulator and the swirler, a Parker metering valve for flow control and Micro-Motion CMF010 Coriolis flow meter for measuring fuel flow are placed and connected by steel-braided hoses. An igniter plug for a P&W PT06 engine is placed on the side wall of the combustor for initial ignition, and a device supplying current energy to the plug, i.e., ignition coil, is a homemade device, as seen in Figure 2.15.

2.3 Experimental Section

Initially, a single pressure transducer was mounted on the combustion rig to measure the pressure oscillations in the combustor. The measured spectral data from this transducer and the microphone outside the combustor, and the pressure oscillation inside the combustion and acoustic emission correlated well. Also, they both correlated well with the measured chemiluminescence data. This proved that the entropy wave phenomena observed by the optical video system coupled with the acoustic signals. The complex details of the variation of measured chemiluminescence by the optical system suggested the use of three additional pressure sensors on the combustor. The purpose of the two pressure sensors in the upstream duct, as shown in Figure 2.3, was to determine the acoustic impedance characteristics at a specified boundary, such as the end of the air heater.

Two different experimental configurations were assembled from the section starting from the air heater and are the subjects for this thesis. The original (short) rig configuration is shown in Figure 2.3. This configuration limited the capability of measurement, explained in a later section; therefore, a new (long) rig configuration was fabricated, as illustrated in Figure 2.4. The (long) rig configuration allows measurement of precise acoustic waves from the combustion region propagating upstream and downstream, by employing ducts of a constant cross sectional area. The other reason for the upgrade is that the original (short) rig configuration allowed only four PCB transducers to be mounted, but the new

(long) rig configuration is capable of mounting ten PCB transducers; the locations and amounts are carefully designed. Pictures of the upstream duct, combustor and downstream duct sections of the (long) rig are shown in Figure 2.10, 2.11, and 2.12, respectively. Additionally, the (long) rig configuration is able to mount a speaker (or speakers) on the duct upstream of the combustor. The diaphragm of the speaker is protected by cooling air flowing through four NPTs mounted on the flange between the experimental rig and the speaker, as seen in Figure 2.13. The speaker is introduced to measure the acoustic impedance at the swirler, the air heater, and the exhaust-duct end to define the boundary

17 characteristics for the experiment by generating tonal or broadband frequency noise. All acoustic signals were acquired simultaneously using a multi-channel data acquisition system described in a later section.

Additional type-K thermocouples, a total of five, were inserted into various sections of the (long) rig to obtain a more accurate temperature profile within the boundaries of interest.

2.4 Instrumentation System

The instrumentation system consists mainly of signal acquisition. In addition, tonal or broadband signal generation is required to excite a speaker (if required). Two full instrumentation wiring diagrams with and without the speaker are illustrated in Figure 2.5 and Figure 2.6. National Instrument LabView computer program code is written to communicate with data acquisition devices and the signal generation device.

Data is saved in Microsoft Excel format and analysis is carried on and completed according to the mathematics introduced in a later section. The pressure transducers utilized are all PCB Model 116B.

These were mounted on sense tubes to protect them from the high temperature air flowing in the test rig.

Further details of the acoustic pressure sensors are discussed in a later section of this thesis. To convert the pressure oscillation phenomena to voltage, each PCB pressure 116B transducer requires an In-Line

Charge and Voltage Amplifier, manufactured by PCB Electronics, i.e., a charge converter. PCB#1 to

PCB#6 illustrated in Figure 2.5 and Figure 2.6 employ charge converter Model 422E51, that has a conversion factor of 1 mV/pC, while CPCB#1 to CPCB#4 use the Model 422D12 with a 10.14 mV/pC factor. The reasoning for utilizing different models is simply because of availability. The conversion difference caused by charge converters are counteracted by a sensor signal conditioner. Two different models of PCB Electronics Line-Type ICP® Sensor Signal Conditioners are employed for instrumentation. Charge converters from PCB#1 to PCB#6 are connected to the signal conditioner Model

483 C Series, which has a gain of x1 for all six channels, while charge converters from CPCB#1 to

CPCB#4 are connected to the signal conditioner Model 482A16 with a gain of x10 for four channels, with the result that all ten PCB 116B transducers have a conversion factor of 100mV/pC. An AKG D112

18 microphone was employed in the test cell to monitor the combustion noise radiating from the exhaust termination of the combustor. The microphone is a condenser microphone; therefore it requires a 48 Volt.

Power for the AKG D112 microphone is supplied by a Samson S-Phantom 48V Power Supply. The microphone is capable of sensing 20Hz to 17,000Hz. Ten channels of pressure oscillation signal from

PCB pressure transducers and one channel of AKG microphone signal lines (BNC cables) are connected to two 24bit, high resolution 8-Channel Dynamic Signal Acquisition Modules, NI PXI-4472 in the PXI

Chassis, and NI PXI-1042, that are capable of communicating with a Windows PC system through a PCI interface module. Type-K thermocouples are inserted into the flow path to measure air temperature and are acquired using an NI-9213, 16-Channel Thermocouple Input Module. The thermocouple input module is capable of communicating with a Windows PC system via USB connection using anNI USB 9162 1-

Slot USB Chassis. Finally, the desired tonal or broadband signal is generated by a Stanford Research

Systems (SRS) Model DS335 3.1MHz Synthesized Function Generator. The communication between a

Windows PC and SRS Model DS335 is established via RS-232. Obtaining this communication is extremely important if the feed-back loop establishment is desired to experiment on relations of the acoustic waves generated by combustion and the in-out phase and various frequency signals that are generated by a speaker. The signal generated by the SRS Model DS335 is amplified by an AudioSource

Amp 5.1, and outputted by a JBL 2446H Horn Driver. This acquisition system is shown in Figure 2.16. In addition to measuring the pressure oscillations in the combustor, chemiluminescence imaging data were also measured to study the flame structure. A quartz window was mounted for flow visualizations at the other side of the stainless steel plate, which contained a grid of PCB transducers. A Phantom V7.3 high- speed CCD video camera was stationed facing toward the combustor window to capture the flame dynamics.

Figure 2.7 Heater Core Picture

Figure 2.8 Schematic of Heater

Figure 2.9 Cooling Air System for a Speaker

Figure 2.10 Upstream Section of New (Long) Combustion Instability Research Rig

Figure 2.11 Combustor Section of New (Long) Combustion Instability Research Rig

Figure 2.12 Downstream Section of New (Long) Combustion Instability Research Rig

Figure 2.13 Cooling Air System for a Speaker

Figure 2.14 Micro Motion Cmf050 Coriolis Flow Meter

Figure 2.15 Homemade Ignition System

Figure 2.16 Data Acquisition System

High Speed Video Data Processing:

Data from the high speed video camera are stored in AVI file format with 800 x 600 pixel resolutions.

Video data requires appropriate conversion from AVI file in order to apply the in-house developed POD or FFT tools, which are both written in National Instrument LabView Version 7.1 software. First, the AVI file format is converted into color (RGB), Red, Green, Blue, Grayscale, or Intensity image, depending on the user‘s choice, then the pixels of an image are extracted to a 2D array of 32 bits (float). The original

800 x 600 pixels are reduced to 200 x 150 cells by taking an average of neighboring 4 x 4 pixels. The current approach has adequate cells to capture the characteristic of combustion dynamics with significantly less computational source requirements. This file conversion loop continues until the last frame of AVI video file, and the 2D array that is converted during the loop is saved in memory, resulting in a 3D array that has a depth of the total frame length, shown in Figure 2.17a. Conversion of the original

3D array from the video file to the new 3D array in form of Fourier Transform is expressed in Figure

2.17a and 2.17b.

Figure 2.17a and Figure 2.17b Three Dimensional Array of Frames to Frequency

PCB Transducer Location and Spacing Between:

Three set of PCB pressure transducers are unevenly mounted on ducts, as illustrated in Figure 2.4.

Introducing uneven space between three PCB transducers is preferred, because one fixed space between transducers will potentially make it difficult or impossible to capture certain frequencies.

PCB Block-Tube:

The tube is continuous from the measurement location to the end of tube termination. The geometrical details and the locations of the two pressure sensors are illustrated in Figure 2.18. The PCB pressure transducer was coupled to a 0.003175m (1/8‖) diameter tube at 0.3048m (12‖) from the location of measurement. The total length of the tube was 7.9248m (25‘+12‖) and the end of tube away from the combustor is terminated by a cap. The tube was continuous over its entire length to minimize any reflection caused by a discontinuity, as shown in Figure 2.18. The length of the tube was chosen because it minimized the effects of the reflected wave from the closed end (due to viscous damping). Thus, for all practical purposes, the sense tube was of infinite length.

Figure 2.18 Schematics of Pressure Sensor on Acoustic Measurements

Because the pressure transducer is not flush-mounted on the wall of the combustion rig, there is a phase and amplitude difference between the sense location (on the combustion rig) and the measurement location of the PCB transducer. To eliminate this problem, each pressure sensor is calibrated relative to a reference pressure transducer, which is flush-mounted at the sense location. An example of the relative calibrations of two pressure sensors is shown in Figure 2.19. (see Appendix III for procedure of calibration)

Figure 2.19 Amplitude and Phase Response of Two Pressure Sensors

Chapter 3: Chemiluminescene – POD and Video FFT (Combustion)

To investigate the connections between acoustic emissions and flame dynamics, the high speed video data for flame energy release rate and acoustic emissions were gathered using a prototype gas turbine combustor sector. Techniques introduced in this section are extremely important to obtain not only the heat release strength, region and flame structure of combustion, but also to obtain the information of the heat-release rate from combustion that relates to Rayleigh‘s criterion, which will be discussed in a later section. Three distinct cases were selected to illustrate the different capabilities of the two considered

(POD and FFT) analysis methods.

Original (Short) Rig: Single Frequency Peak:

Single frequency peak combustion was observed with the original (short) rig configuration, as shown in the schematic in Figure 2.2. A radiating acoustic wave, recorded by an AKG microphone from the experimental combustor with particular test conditions and fuel, showed a dominant single frequency band at 280 Hz, as shown in Figure 3.2. A single frame of high speed video is shown in Figure 3.1. The time-average mean and rms chemiluminescence intensity from 3000 video frames are shown in Figures

3.3 and 3.4. The peak of flame dynamics occurs at the swirling flames inside the combustor dome (red color in Figure 3.4).

New (Long) Rig: High Fuel/Air Ratio (f/a=0.0343) and Low Fuel/Air Ratio (f/a=0.008):

Two different fuel/air ratio cases with propane as fuel were selected to illustrate the investigation technique and discuss the result from the experiment with the newly designed (long) combustion rig. The results obtained from those two test conditions with the newly configured experimental rig are shown in the schematic in Figure 2.3. The two test cases gave the researcher two distinguishable results. Radiating acoustic waves were recorded by the AKG microphone stationed near the experiment rig. The first case

(Case 2) is the high fuel-to-air ratio case, and a microphone showed that the rig emitted a low amplitude

33 wideband frequency acoustic wave, as seen in Figure 3.19. The second case (Case 3) is the low fuel-to-air ratio case, which had several high amplitude narrow band frequencies in the radiating acoustic wave from the rig (Figure 3.19). A quick glance of the FFT temporal plot from the microphone in Case 3 shows that combustion instability is occurring during the combustion process. The difference of amplitudes between those two cases is quite obvious in Figure 3.19, and supports the argument that combustion instability occurs in Case 3.

To eliminate confusion, test conditions and set ups were assigned case numbers.

Case 1: Single frequency peak from the original rig.

Case 2: High fuel-to-air ratio tested on the newly designed rig.

Case3: Low fuel-to-air ratio tested on the newly designed rig.

3.1 Theoretical Notes

Let us consider a function Z(x,y,t) as combustion dynamics captured by high speed video camera.

Freedom in the x and y direction represent spatial coordinates, and pixel number in the video, and t direction is the temporal coordinate, or frame number.

The Proper Orthogonal Decomposition:

The Proper Orthogonal Decomposition is a technique used for separating a signal into spatial and temporal functions. Reconstruction of the original signal via separated functions is also possible. The principal characteristics to be determined by POD are mode shape , and its corresponding waveform , of the high speed video.

 tyxztyxZtyxZ ),,(),,(),,( (3-1)

Summation of the variable separated form in equation (3-2) is complete if M approaches infinity, but function z(x,y,t) is finite by limiting the frame number for our application; therefore exact reconstruction/decomposition is possible.

M   nn yxtatyxz ),()(),,( n1 (3-2)

The expression in equation (3-2) is called Proper Orthogonal Decomposition (POD). We must note that spatial functions are orthonormal to each other given the equation (3-3), where D is the spatial domain to solve equation (3-2) efficiently.

 if  nn 21 )(1 n  n ),(),( dxdyyxyx   D 1 2 0 otherwise  (3-3)

Spatial function , can be found by determining eigenfunctions. This technique is called the Method of Snapshots, referred by Sirovich [2]. This problem can be solved if the data contain equally spaced time intervals. Indeed, our high speed video data fits these criteria. Superscripts i and j represent the frame number.

1 i)( j)( C)( ij  ),(),( dxdyyxvyxv D M 1, ,..., Mji (3-4)

M is the total frame number of data, and M x M matrix C is constructed by equation (3-4) using frame by frame high speed video. The relationship of eigenfunctions and eigenvectors is expressed in equation (3-

5). The magnitude of corresponding eigenvalues to modes represents the energy of its POD modes.

CA n)(   A n)( n 1,..., Mn (3-5)

Substituting the eigenfunctions defined from equation (3-5) into equation (3-6), we are able to define spatial function , which is the POD mode shape. Note that eigenfunctions are a linear combination of high speed video data.

M kn )()( n ),(   k yxvAyx ),( k 1 (3-6)

th Waveform , which corresponds with the POD n mode shape, is expressed by equation (3-7). Keep in mind, orthonormality in spatial function, , temporal coefficient, , that only depends on

is solved.

n  n xtxzta )(),()( dx D (3-7)

The Fourier Transformation:

The DC component is taken away, as shown in equation (3-1). The AC component in Fourier

Transformation is expressed in equation (3-8).

M nti   n ),(),,( eyxbtyxz n1 (3-8)

 titi  if  mn )(1 ee mn dt   D 0 otherwise  (3-9) bn(x,y) is a complex number representing the amplitude and phase angle associated with a specific frequency at a specific spatial location. In Fourier Transformation, the temporal functions are orthonormal to each other. The form resembles the equation (3-3), which expressed orthonormality in spatial function.

3.2 FFT Analysis

The temporal FFT, which is based on the finite number of video frames (3000), has a frequency resolution of 2.2 Hz for the single frequency band with the original (short) experimental rig (Case 1), and a frequency resolution of 1 Hz for the two different fuel-to-air mixture rate results from the newly redesigned (long) experimental rig (Cases 2 and 3).

Case 1:

The temporal FFT spectrum based on the total integrated intensity (sum of all pixel intensity) of each frame was first conducted in Case 1. The resulting energy release amplitude spectrum, as shown in Figure

3.5, is similar to the acoustic spectrum in Figure 3.2 and is dominated by a single frequency band around

279.5 Hz. This indicates the acoustic emissions and energy release dynamics are closely related. The spatially distributed FFT amplitudes from Case 1 at three neighboring frequencies, around 280 Hz (277.3,

279.5 and 281.7) are shown in Figure 3.6. The amplitude levels at 277.3 and 281.7 Hz are significantly lower than those at 279.5 Hz, indicating extremely narrow band flame oscillations at 279.5 Hz. The spatially distributed phase angle (radian) contours associated with 279.5 Hz are shown in Figure 3.7.

From Figures 3.6 and Figure 3.7, the 279.5 Hz flame dynamics are concentrated in three amplitude local maximums. Two local maximum spots (45, 100) and (55, 25) are near the swirler exit with much higher amplitudes than the third local maximum spot (80, 65), at the centerline downstream. Also, the two spots with high amplitude coincidence of the highest flame dynamics spots of the r.m.s. and mean amplitude contours are shown in Figure 3.3 and Figure 3.4, indicating the acoustic energy source is the flame front oscillations. The majority of phase angles at the high amplitude regions have values close to 3.14 or -3.14 radians, indicating the majority of energy releases from the flame are in phase. However, there is a local maximum phase angle of around 0 radians located at the lower swirler exit area (55, 25); indicating some energy release rate from the flame front is out of phase with the main flame dynamics. Actually, two local extreme spots, (55, 25) and (45, 100) near the swirler exit have a phase angle difference of 180 degrees, which provides clues that a small circumferential energy release dynamic may be the trigger for the main combustion dynamics.

Case 2: f/a=0.0343

The temporal FFT frequency domain spectrum obtained from the AKG microphone showed that no narrow band high amplitude frequencies exist in acoustic waves emitted from the combustion during the

Case 2 experiment (Figure 3.19). All the frequencies up to 1000Hz in the FFT spatially distributed domain plots are examined and three were determined as significant, or plots that contain high amplitudes among other frequencies (106Hz, 208Hz, and 354Hz), were selected (Figure 3.20). Note that the frequency of 208 Hz is near the harmonic frequency of 106Hz. The location of low-frequency amplitude peaks, 106Hz and 208Hz, are concentrated at 1/3 length to 1/2 length of the combustor. Amplitudes of

37 peak regions (100, 100) and (40, 100) for 106Hz, and (100, 100) for 208Hz, are also low, while spatially distributed in a large region in the combustor. The third frequency peak, 354Hz, shows a different shape when compared to the first two frequencies of 106 and 208Hz in the spatial domain FFT plot. Streaks of lines are visible from (20, 120) to (60, 110) below the flow path center line and (110, 135) to (80, 115) above the center line. Those stream lines go along with the dilution-hole jet stream paths and distinctive non-harmonic frequency as shown in Figure 3.20. The dilution holes are apparently responsible for this

354Hz, and it does not appear in the Case 3 microphone FFT spectrum or spatially distributed FFT domain plots.

The spatially distributed phase angle plots of 106Hz and 208Hz from Case 2 (Figure 3.20), show that the phase angle of 3.14 is a location of high amplitude regions, which concludes that the energy release rates are in phase. Unlike the result obtained from the spatially distributed phase plot in Case 1, there is no 180 degree phase difference crossing the flow path centerline, which indicates that there are no spinning motions associated with these two frequencies. The third notable frequency, 354Hz oscillation, is potentially caused by a dilution-hole jet stream, which was mentioned earlier. The spatial phase plot of the third frequency, 354Hz, from Case 2, shows the phase angle of 3.14 to -3.14 from the dilution-hole areas into the combustor downstream main flow path (toward the centerline). The cycles of heat release on the path along dilution hole jet streams are visible in the phase plot from the (30, 120) bottom dilution hole, and the (90, 125) top dilution hole.

Case 3: f/a=0.008

Numbers of notable high amplitude narrow band frequencies were emitted during combustion from the

Case 3 experiment as shown in the temporal FFT spectrum in Figure 3.19. The highest amplitude obtained from an AKG microphone was five times of the peak amplitude obtained from the Case 2 experiment result. Four distinctive plots showed significantly high amplitudes by the scanning of FFT spatially distributed amplitude plots from 1Hz to 1000Hz and selected for the investigation. The

38 frequencies associated with those four distinctive plots are, 106, 214, 322, and 428Hz as shown in Figure

3.24, which correlate to peaks obtained from the microphone FFT spectrum shown in Figure 3.19. The peak regions of FFT spatially distributed plots are all near the swirler; two maximum zones (90, 150) and

(20, 140) are visible in plots. Additionally, all four FFT spatial plots show the amplitudes being concentrated on the upstream side of the 1/2 length of the combustor, which indicates that reactions are occurring near the swirler or within the swirler cup region.

The spatial phase angle plots show plane-wave-like heat release from upstream to downstream in all four frequency plots. Unlike Case 1, there is no indication that a spinning motion exists during the combustion process of Case 3 by investigating the four selected spatially distributed phase plots. Another notable contrast is that the effect of dilution hole jet streams was visible in Case 2, but there is no contribution of dilution hole jet streams seen in the spatially distributed FFT amplitude or the phase angle plots in any of the four major frequency peaks in Case 3, which indicates that the distinct high amplitude frequencies appearing in the temporal FFT spectrum, as well as the spatially distributed FFT plots, are potentially harmonics of 106Hz.

Reconstruction Demonstration with Case 1:

Based on the amplitude and phase contours (Figures 3.6 and 3.7), we can reconstruct the motions of the energy release rate associated with 279.5 Hz acoustics. Figure 3.8 illustrates one complete cycle of 279.5

Hz dynamics: 12 plots with a 30 degree phase angle increment. The physical time step associated with a

30 degree phase angle is about 0.3 ms (1/279.5*30/360 second). The illustrated motion is essentially the averaged (279.5 Hz) cyclic motions based on the total number of cycles (about 120 cycles in the current study) of the video recordings. The variations among cycles tend to be washed out in Figure 3.8. As discussed before, the majority of energy release in the whole combustor is, in phase, dominated by the same color, either blue (negative values) or brown (positive values), in the individual phase angle plot.

However, there are mixed colors (positive and negative values) in the plots with phase angles of 90, 120,

270 and 300 degrees. The amplitude levels in those four plots are much less than that in other phase angles. Furthermore, the spatial structures are similar among these four plots, but the color is reversed between plots with a 180 degree phase angle difference. The above observations support the physical phenomena that a ―small‖ circumferential energy release dynamic co-exists with a ―big‖ in-phase energy release dynamic at 279.5 Hz. The wavelength of 279.5 Hz acoustics is significantly longer than the current spatial domain captured by high speed video. If positive feedback exists between acoustics and energy release (thermal acoustic instability), the resulting high amplitude energy release should be in- phase. Again, it is concluded that circumferential energy release dynamics may be the trigger for the main combustion dynamics. Figure 3.9 shows roughly one cycle (279.5 Hz) of original video on the dynamics part of the energy release rate. Plots are increased by two video frames, which corresponds to a 30.5 phase angle at 279.5 Hz. It is extremely hard to visualize the spatial structure associated with 279.5 Hz cyclic behavior. Motion plots, as seen in Figure 3.8, based on FFT amplitudes and phase angles at 279.5

Hz, essentially act as the ―best‖ narrow band filter at 279.5 Hz to isolate the motion associated with this frequency.

3.3 POD Analysis

This technique (POD) is optimized so that it will analyze the largest amount of system energy with the least (spatial) modes of any system dynamics decomposition, which is a suitable method for describing/modeling a complicated energy release structure. Therefore, using POD will enable us to create a few (spatial) mode shapes, which contain the majority of energy of the analyzed dynamic system, and the reduced dynamic models can be more easily established. The temporal behavior associated with each mode may contain different frequencies and phase angles, therefore making it possible to pinpoint which (spatial) mode shapes are associated with frequencies of interest. In this effort, we will show that

POD is an effective spatial filtering tool for the application examined.

Case 1:

The first three spatial mode shapes and their temporal spectra are plotted in Figures 3.10 - 3.12. All modes shown here have different shapes, but their corresponding time-domain waveforms have the same frequency (279.5 Hz). Samples of time-domain waveforms are illustrated in Figure 3.13, and it is evident that small phase differences exist between modes even though they oscillate in same frequency. Mode energies are calculated while solving equations (5) which are the corresponding eigenvalues to the modes.

Accumulated energy up to the first 125 modes is shown in Figure 3.14. In the current analysis, 3,000 spatial modes and the associated energy have been resolved, since 3,000 video frames are considered in the POD analysis. The first 125 modes contain 85% of the energy total of 3,000 modes, and the first three modes contain 19%, 7% and 5% of the energy, respectively. The evidence that POD analysis can use the

―minimum‖ number of spatial modes to represent the dynamic behavior is validated.

In the POD analysis, spatial modes cannot have the same frequency without a phase shift, otherwise, these spatial modes can be combined into a single spatial mode with higher energy. In the FFT analysis, the phase angles are dependent on the spatial locations—different phase angles at different locations.

However, in the POD analysis, each spatial mode has one temporal waveform, so the whole mode oscillates as a unit modulated with the associated temporal waveform. If the waveform is dominated by a single frequency, positively valued regions and negatively valued regions in the spatial mode shape will oscillate with a 180 degree phase angle difference.

The POD mode #1 structure is very similar to the amplitude contour plots of FFT (Figure 3.6). Three local maximums and minimums, (55, 25), (45, 100) and (80, 65), are in the same locations as the local maximums in the amplitude plots of FFT. Maximum (positive values) and minimum (negative values) spots in the POD, as discussed, have a phase angle difference of 180 degrees, which is also calculated by the phase angles of FFT in Figure 3.7. It can be concluded that the POD mode #1 has extremely similar

41 dynamics, in both temporal and spatial behavior, as those isolated by the FFT filtered at 279.5 Hz. The

POD analysis also predicts that the 18% of dynamic energy is associated with this mode.

Case 2: f/a=0.0343

The first three POD modes from Case 2 and their corresponding time-domain frequency plots are shown in Figure 3.21. The first glance at all three mode shapes indicates that they are complicated and do not show peaks in large regions. Temporal FFT spectrums that correspond to individual modes indicate that all three modes have a frequency of 106Hz, but local oscillation regions have small amplitudes while widely spread and distributed within the combustion region. By looking closely at the POD mode #1, a trace of the dilution hole jet stream line is visible as a part of the mode shape. The heat release associated with the dilution hole jet stream was defined as having a 354Hz oscillation in the earlier discussion of spatial FFT analysis, having regions of stream toward the flow path centerline being visible from (20,

120) to (60, 110) below the flow path center line and (110, 135) to (80, 115) above the center line.

Likewise, POD mode #1 not only indicates the region, but also, the temporal spectrum indicates the

354Hz. Figure 3.22 shows the time-domain waveform of the first three modes of Case 2, and phase differences are evident. As was mentioned in Case 1, POD modes with the same frequency can co-exist in the same dynamics as long as a phase angle difference exists. Total POD mode energy in percentage is shown in the plot in Figure 3.23. Notice that the first mode contributed below 5%, which states that the significance of this mode as well as that of the second and third modes is small; therefore it takes 300 modes to meet 85% of the total modes. Observations of POD mode shapes and mode energies explain even-heat-release in combustor regions by mode shape, while the most dominant mode shape contributes a small amount as a whole, which is indicated by the accumulated mode energy plot in Figure 3.23.

Case 3: f/a=0.008

As expected from the result of the spatially distributed FFT plots, the first three POD mode shapes from the Case 3 experiment showed a much more simple shape, with the dominant peak region being larger than the POD modes of Case 2. POD mode #1 has a similar spatial distribution to the FFT amplitude plots as well as distinctive frequency peaks. The heat release regions of Case 3 are concentrated near the exit of the swirler, which agrees with the result discussed in the earlier section of the spatially distributed FFT

43 amplitude plots. POD mode #1 has the shape of a swirler cup; with that in mind, POD mode #2 has the peak region (60, 160), that is similar to the missing portion of the swirler cup region in mode #3 (60,

160). This idea points out that mode #2 and mode #3 are potentially related to a singular main motion, which is mode #1. Those two modes have the same frequency, but have different phases; it is explained earlier that they can co-exist in the same dynamics. The POD modes‘ corresponding time-domain frequency plots show much higher amplitudes than in Case 2, which tells us that the Case 3 POD modes have much stronger oscillation than the modes in Case 2. Time-domain wave forms of POD modes are shown in Figure 3.26. Mode #1 and mode #3 have a similar wave form with a small phase delay, while mode #2 has a significantly different form, which can be explained by the accumulated POD mode energy plot in Figure 3.27. Notice that mode #1 contains more than 65% of the total energy, and mode #2 contributes 10% of the total energy while mode #3 indicates 3.5% of the total energy. Therefore, mode #1 is the absolutely dominant mode in these combustion dynamics, and mode #3, which has a similar spatial distribution (mode shape) as that of mode #1 while the time-domain wave form with the small phase delay contributes a small portion when compared to mode #1.

Reconstruction Demonstration with Case 1:

POD mode #1 and its corresponding time-domain waveform is reconstructed and shown in Figure 3.15.

Plots are increased by two video frames, which corresponds to the 30.5 phase angle at 279.5 Hz, and the

12 plots present almost a complete cycle. Since the time domain is dominated by a single frequency, the locations of local maximums and minimums are not changed with time or phase angle. This is significantly different from the reconstruction generated by the FFT analysis. Figure 3.8 shows that the local extremes move with phase angles or time since the local amplitude is modulated by a difference phase angle and a higher spatial frequency structure exists at different phase angle plots. The

―continuous‖ motion and higher frequency structure cannot be captured by a single POD mode with its associated time-domain waveform. The supplemental information will be provided by the higher POD modes, which contain less individual mode energy.

The POD modes #2 and #3 contain more local extremes with a shorter separation distance (higher spatial frequency) compared to the POD mode #1. The energy associated with the #2 and #3 modes are 7% and

5%, respectively, which is significantly less than the 18% of mode #1. Local extremes contained in modes

#1 and #2 show up at FFT reconstructed plots (Figure 3.8) at different phase angles, but theses extremes have less amplitude compared to the peaks identified by mode #1. This result also verifies the statement that POD is a suitable tool for modeling complicated phenomena such as fluid motion and/or combustion dynamics, because it captures/describes high dimensional data with minimum information. At the same time, it allows researchers find characteristics of dynamics phenomena.

The Wave FFT 18000

16000

14000

12000 10000

8000 Magnitude

6000

4000 2000 0 0 100 200 300 400 500 600 700 800 900 1000 Frequency in Hz Figure 3.2: Acoustic Emission Spectrum Figure 3.1: A Frame of High Speed Video

Figure 3.5: Energy Release Spectrum

Figure 3.3: Mean of Energy Release Figure 3.4: r.m.s. of Energy Release

Figure 3.6: Amplitude Contours of Energy Release at 277.3, 279.5, and 281.7 Hz

Figure 3.7: Phases Angle (radian) Contours at 279.5 Hz

Figure 3.8: One Cycle of Energy Release Motion at 279.5 Hz (12 Plots with 30° Increments)

Figure 3.9: (Approximate) One 279.5 Hz Cycle of Original Video on Fluctuating Energy Release Component (Start with Frame 200 with Frame Increment of 2 or Phase Angle Increment of 30.5 Degrees)

Figure 3.10: POD Mode #1 Contours and Associated Temporal Spectrum

Figure 3.11: POD Mode #2 Contours and Associated Temporal Spectrum

Figure 3.12: POD Mode #3 Contours and Associated Temporal Spectrum

Figure 3.13 POD Temporal Coefficients as a Function of Time

Figure 3.14 POD Accumulated Mode Energy

Figure 3.15: (Approximate) One 279.5 Hz Cycle of POD Mode #1 Reconstruction (Start with Frame 200 with Frame Increment of 2 or Phase Angle Increment of 30.5 Degrees)

Figure 3.16: (Approximate) One 279.5 Hz Cycle of POD Mode #2 Reconstruction (Start with Frame 200 with Frame Increment of 2 or Phase Angle Increment of 30.5 Degrees)

Figure 3.17: (Approximate) One 279.5 Hz Cycle of POD Mode #2 Reconstruction (Start with Frame 200 with Frame Increment of 2 or Phase Angle Increment of 30.5 degrees)

Figure 3.18: (Approximate) One 279.5 Hz Cycle of Sum of First Five POD Reconstruction (Start with Frame 200 with Frame Increment of 2 or Phase Angle Increment of 30.5 degrees)

Results from the new rig:

Figure 3.19: Acoustic Emission Spectrum from Case 2 and Case 3

Figure 3.20: Amplitude and Phase Contours of Energy Release Rate at 106, 208, and 354Hz, Case 2

Figure 3.21: POD Modes #1, #2, and #3 Contours and Associated Temporal Spectrum, Case 2

Figure 3.22: POD Temporal Coefficients as a Function of Time, Case 2

Figure 3.23: POD Accumulated Mode Energy, Case 2

Figure 3.24: Amplitude and Phase Contours of Energy Release Rate at 106, 214, 322, and 428Hz, Case 3

Figure 3.25: POD Modes #1, #2, and #3 Contours and Associated Temporal Spectrum, Case 3

Figure 3.26: POD Temporal Coefficients as a Function of Time, Case 3

Figure 3.27: POD Accumulated Mode Energy, Case 3

Chapter 4: Acoustic Wave (FFT and Impedance)

4.1 Test Results and Discussion – Original (Short) Rig Configuration: FFT

Figure 4.1 Power Spectrums of Microphone and Four PCB Pressure Transducers

The pressure oscillations, the acoustic emissions, and the energy release dynamics of the combustor sector under investigation were surveyed over a wide range of equivalence ratios, air inlet temperatures, air pressure drops, and fuel types on the original (short) combustor research rig.

Although the amplitude and frequency of acoustic emissions varied with different test conditions, their general behavior and trends were closely tied with the overall air/fuel equivalence ratios. In this section, two test cases (equivalence ratios of 0.14 and 0.27) were selected for discussion of the characteristics as well as limitations of the rig configuration.

Figure 4.1 shows the acoustic and pressure power spectra from five sensors in test Case 1a (equivalence ratio of 0.14). The AKG microphone, which is located outside of the combustor test rig, measured a sharp peak at 273.4 Hz as well as a peak at 442.8Hz. Note that the spectral data measured by the microphone is purely due to acoustic waves radiating from the exhaust plane of the combustor. Therefore, these two frequencies are of principal interest to us in this Case 1a. The pressure sensors PCB1 and PCB2 are mounted in the upstream duct, and PCB3 and PCB4 are mounted on the combustor. Plane waves generated in the combustor propagate through the swirler into the upstream ducts in the test rig. Pressure transducers PCB3 and PCB4 in the combustor record the 273.4Hz and 442.8Hz peaks clearly, as expected, but upstream transducers PCB1 and PCB2 show only the lower frequency peak, which is 273.4

Hz. This means that, for some reason, the acoustic waves corresponding to the peak at 442.8 Hz did not propagate upstream through the swirler. Moreover, the level of this peak is much lower at the location of transducer PCB4.

Figure 4.2 Energy Release Rate at 273.4 Hz, Case 1a

The reasoning for this phenomenon may be discussed by examining the data obtained with the chemiluminescence imaging method at these two frequencies as shown in Figures 5.2 and 5.3. Because of the low equivalence ratio, the combustion energy release only occurs in the swirling jet flows within the dome. Note that the region of maximum intensity of luminescence is roughly the same size and shape at both the frequencies (273.4 and 442.8 Hz.). However, the variation of the luminescence intensity at the lower frequency is nearly symmetrical about the centerline of the swirler. This means that the entropy wave source generated plane acoustic waves and symmetrical acoustic modes (in the transverse direction). At the higher frequency of 442.8 Hz, the variation of the chemiluminescence intensity is asymmetric. This indicates that the entropy wave source generated mainly asymmetric acoustic modes.

The transverse (normal to the direction of flow) dimensions of the combustor are too small to allow the unattenuated propagation of any transverse acoustic modes, symmetric or asymmetric, which may be generated in the source region of the flame. Such acoustic modes are ―cut off‖ and are highly attenuated, or damped. We believe that the principal acoustic mode at 442.8 Hz is the first asymmetric acoustic mode generated in the flame region. It is cut off and therefore, it cannot propagate upstream. In the downstream

66 direction, the high damping associated with this mode is demonstrated in the significant reduction of amplitude between transducers PCB3 and PCB4. Its energy radiates outside to the microphone only because of the relatively short length of the combustor. Note that the combustor is not long enough to justify the ¼ wave resonance at 442.8 Hz. The ¼ wave resonance would occur at approximately 650 Hz, as indicated by the low level acoustic emissions in Figure 1.1.

In light of the above discussion, we believe that the only true resonance in the combustion rig occurs at the lower frequency of 273.4 Hz. At this frequency, a half acoustic wavelength fits (approximately) between the heater and the swirler. We also believe that the combustor contains 442.8 Hz unsteady energy release, but it did not interact with the duct acoustics. If the test geometry varies (single cup to multiple cups or full annular combustor) or the boundary conditions at upstream and downstream ends change

(compressor and turbine), the combustion system may also generate resonant combustion instability at this frequency.

Figure 4.3 Energy Release Rate at 442.8 Hz, Case 1a

Figure 4.4 Power Spectrum of Microphone and Four Pressure Sensors

Figure 4.4 shows the acoustic pressure power spectra from five sensors in test Case 1b (equivalence ratio of 0.27). As in Case 1a, the pressure sensors PCB1 and PCB2 are mounted in the upstream duct, and

PCB3 and PCB4 are mounted on the combustor. The AKG microphone is located outside of the combustor test rig. Pressure transducers on the combustor show two frequencies that are measured by

AKG microphone, but PCB3 shows an additional peak at 525 Hz that is not significant in other transducers.

The intensity of chemiluminescence data measured at 282.8 Hz is shown in Figure 4.5. The source region is not as compact as in Case 1a, indicating a more dispersed combustion mode for intermediate equivalence ratios. Moreover, intensity variation is very asymmetric, indicating generation of several asymmetric transverse modes and plane wave generation at a fairly low level. Consequently, at this low frequency, the acoustic levels measured by PCB1 and PCB2 in the upstream ducts are very low. The transverse acoustic modes were damped as they propagated downstream. However, the acoustic radiation from the exhaust plane at these low frequencies was very efficient, as shown in the spectral data from the microphone.

Figure 4.5 Energy Release Rate at 282.8 Hz, Case 1b

The second peak at approximately 425 Hz is very pronounced at the location of PCB3 in the combustor.

Again, it appears that this is due to a highly damped transverse mode generated in the flame region. It attenuates rapidly as it propagates downstream, as shown in the spectrum for PCB4. It does not propagate upstream but does radiate to the microphone. The third peak at 525 Hz, as shown in the spectrum for

PCB3, is due to a highly damped transverse mode generated in the flame. Its damping rate is so high that it was not detected by PCB4 or the external microphone.

We conclude that the only true resonance in the combustion rig occurs at the lower frequency of 282.8 Hz

(evidenced by strong heat release amplitude and high upstream dynamic pressure amplitude). The unsteady energy release rate at 424 Hz (supported by video and pressure oscillation inside the combustor) did not interact with combustor acoustics (low amplitude in upstream ducts). The 525 Hz measured with the combustor is only the local hydrodynamics instability (no energy release rate peak and no upstream pressure peak).

In conclusion, the only evidence of a combustion driven acoustic resonance is at the lower frequencies

273.4 Hz and 283 Hz. All the other acoustic data indicate the generation of highly damped transverse acoustic modes in the flame region.

4.2 Conclusion of Test Result from the Original (Short) Rig Configuration

A careful examination of the measured acoustic data and the corresponding chemiluminescence imaging data shows that the only true acoustic resonance occurred during the fuel-lean operation of Case 1a. It occurred at 273.4Hz. This resonance had the potential for causing combustion instability. All other high amplitude pressure oscillations (at higher frequencies) were only measured in the flame region inside the combustor. They were highly damped and did not propagate into the upstream duct and were attenuated as they propagated through the exhaust termination of the combustor. It is concluded that these high

70 amplitude oscillations in the combustor were associated with transverse acoustic modes that were generated in the flame. These modes were acoustically cut off and therefore, were highly damped.

4.3 Suggested Changes in the Original (Short) Rig Configuration

The acoustic field within the combustor was considered too complex to describe accurately from the measured acoustic data; for example, PCB transducers on the combustor wall were placed in series along the centerline (flow path) which could not identify circulation motion. Therefore, further investigation of combustion instability phenomena should be carried out in a modified combustion rig.

The following changes in the design of the rig were made:

(i) Constant cross sectional area duct segments were incorporated upstream and downstream of the

combustor.

(ii) Additional pressure transducers were installed in these new duct segments. The objective of these

transducers is to measure the amplitudes of the forward and backward propagating plane waves in

the duct segments of the new apparatus.

4.4 Results and Discussion – New Rig (Long) Configuration FFT

Two distinct cases, high fuel-to-air ratio (Case 2) and low fuel-to-air ratio (Case 3), on the newly configured combustion rig were chosen for discussion. (Note* Detailed description of the two selected distinct cases are expressed in the Chapter 3, Section 3.3 POD Analysis) PCB pressure transducers were placed on selected locations of the newly configured combustor test rig, as shown in Figure 2.2. Four

PCB transducers were placed on the combustor wall as shown in Figure 4.6. The locations were determined in order to measure potential transverse direction wave propagation, or circulation, if there is any. The original (short) combustor test rig was not capable of this, because the PCB transducers on the wall were lined on the centerline flow path.

Figure4.6: PCB Transducers on the Combustor Wall

As in the previous experiment with the original (short) combustor rig, the AKG microphone was placed outside of the combustor test rig. Despite all the in-duct PCB transducers showing 106Hz in their power spectrum, there is no 106Hz spike shown in AKG microphone spectra (Figure 4.7). This could be explained by the location of the microphone, the exhaust duct termination, or the radiation impedance which may have high resistance close to 106Hz. Thus it acts like a hard wall termination and prevents the radiation of sound at 106Hz. Radiation impedance of the exhaust duct is discussed later in the section of

Impedance Calculation. PCB #1 is the upstream transducer further away from the combustor. PCB#4 is on the exhaust duct, and the orientation of the PCB transducers in the combustor is shown in Figure 2.2.

The attenuation effect of the swirler can be seen by examining the amplitude drop in the power spectrum of PCB#1. While 354Hz is clearly visible in the power spectrum of all PCB transducers, 206Hz is difficult to observe in the PCB#1 plot. The answer for this attenuation characteristic is in the impedance of the swirler, which will be discussed in a later section. Unlike the results obtained in Case 1a and Case

1b from the original (short) combustor rig, which showed non-identical frequency and peak amplitudes,

Case 2 shows visually identical frequencies and amplitudes from all the PCB transducers mounted on the combustor wall. This indicates that the wave propagation is symmetric along the flow direction centerline.

In Chapter 3, chemiluminescene investigation of Case 2 concluded that there is no spinning motion, i.e., no asymmetrical modes, and the PCB transducers mounted on the combustor wall tell the same story.

Figure 4.7 Power Spectra of the Microphone and the PCB Transducers, Case 2

Figure 4.8 shows the acoustic and pressure power spectra for the test Case 3 (lower fuel-to-air ratio) from the new (long) combustor rig. Strong discrete (tonal) frequency peaks are clearly visible in all the plots.

The AKG microphone is not picking up a 106Hz frequency peak as it was in Case 2. The positioning of the microphone and/or acoustic wave attenuation due to the exhaust duct radiating impedance are believed to be the cause for the missing 106Hz peak from the AKG microphone spectrum. Similar to Case

2, the spectrum of PCB#1 shows that the amplitude of 206Hz and well as 106Hz are attenuated by the

73 swirler. All four PCB transducers on the combustor wall show visually identical spectrums. This indicates that the acoustic wave is a plane wave, with symmetric modes in the transverse direction (along the flow path centerline). The characteristics of plane waves in Case 3 are observed by chemiluminescene analysis, which is discussed in Chapter 3. The frequency peaks seemed to be harmonic to 106Hz, and amplitude goes down as frequency increases. The y-axis (amplitude) of the PCB power spectrum is set low in Figure

4.8 to show other existing frequency peaks, but Figure 4.9 is full-scale plots, which shows how the 106Hz peak is the dominant peak.

Figure 4.8 Power Spectrum of Microphone and PCB Transducers, Zoomed In, Case 3

Figure 4.9 Power Spectrum of Microphone and PCB Transducers, Full Scale, Case 3

4.5 Pressure Dynamics Measurement

4.5.1 Acoustic Transmission and Impedance: Step Area Change and Gradual Area

Change

Figure 4.10: Schematic of Upstream Duct and Combustor

The 0.154m (6.065 inch) inner diameter tube with a length of 1.143m (45 inches) was placed upstream of the SAS combustor. Three PCB transducers were employed to measure swirler impedance, location xSU, from upstream as well as impedance of air-heater at x0 by changing the speaker location to the left downstream side of the transducers. Upstream PCBs were responsible for measuring swirler impedance

(ZSU) and air-heater impedance (Z0). At x1, the propagated acoustic wave in the longitudinal direction faced step change (constriction) from a 0.154m (6.065 inch) diameter tube into an SAS diffuser inlet area of 2.961x10-3 m2. Propagating acoustic waves experienced gradually changing cross-sectional areas from x1D to xSU. The locations of the step area changed and gradual area changes were seen in Figure 4.10.

Acoustic damping caused by friction of the rig wall and between air particles exists in real life, but throughout the acoustic wave measurement expressed in this thesis, the effect of damping is considered to be small and treated as negligible, therefore it is not considered.

Acoustic Pressure and Particle Velocity:

Acoustic pressure and acoustic particle velocity of plane waves at any location within the combustor rig boundary can be expressed with complex acoustic wave amplitudes (forward and backward going propagating waves) as:

Acoustic Pressure at any location xm:

(4-1)

Acoustic Particle Velocity at any location xm:

(4-2)

Where

A: Incident complex acoustic amplitude (forward going propagating plane wave amplitude)

B: Reflected complex acoustic amplitude (backward going propagating plane wave amplitude)

k: Wave number,

f: Acoustic F\frequency c: Speed of sound i: imaginary number

Figure 4.11 Step Area Change

Step Area Change:

Two necessary conditions that were needed to be satisfied for step change (Fig 4.11) are:

1) Continuity of Acoustic Pressure

2) Continuity of Volume Displacement (Conservation of Mass)

The equations corresponding to the above conditions at the boundary x1 are:

(4-3)

(4-4)

And ―D‖ represents downstream side, ―U‖ represents upstream side

(4-5)

Where (4-6)

(4-7)

Using the relation expressed in equation (4-5), the impedance of downstream side at x1 is obtained.

(4-8)

Diffuser Section Area Change:

Figure 4.12 Gradual Area Change

Our diffuser had three segments. Unlike the step change that propagating wave experienced at the flange, the diffuser had gradual area change, as shown in Figure 4.12.

Gradual Area Change:

Impedance at the beginning of the diffuser was defined by using equation (4-8), and area is constant up to section X2.

Introducing reflection coefficient R at x1D in the equation and letting x1D = 0, then:

(4-9)

(4-10)

From section x2 to x3, the propagating acoustic wave experiences gradually changed across sectional areas. This involved the assumption that there is no energy loss from point x2 to x3, which is expressed in terms of transmission coefficient to be 1.

(4-11)

Therefore

(4-12)

Since the area change was involved, it is be suitable to use the term Acoustic Impedance, Z,

(pressure/volume velocity) instead of Specific Acoustic Impedance, z, (pressure/particle velocity).

Acoustic Impedance = (4-13)

Therefore

(4-14)

Specific Acoustic Impedance, z, is calculated by using complex acoustic wave amplitude:

(4-15)

Also, pressure and particle velocity at x3, after the propagating acoustic wave went through a gradually changing cross sectional area is expressed as follows:

(4-16)

(4-17)

Where

Dividing Equation (4-17) by Equation (4-18):

(4-18)

Square root term introduced in equation (4-19) cancels out:

(4-19)

Note* Equation (4-16) and (4-17) are expressions of acoustic pressure and particle velocity in a gradually changing cross sectional area. Since acoustic impedance is defined by division of acoustic pressure and particle velocity, the area ratio in the square root term cancels out. Therefore, identifying acoustic impedance from point xm to xn within gradually changing area ducts only depends on the final local cross sectional area (where the impedance is defined) as shown in equation (4-19). This is the result of the first approximation. However, the laboratory experiment result showed sufficient correlation between the theory using equation (4-19) and the measurement obtained by pressure transducers. (See Appendix I)

Therefore, the new location of impedance through a gradually changing cross sectional area duct from location x1 to x2 is defined by using equation (4-18):

(4-20)

Introducing reflection coefficient R2 at x2 in the equation and letting x2 = 0, then:

(4-21)

Now the impedance relationship between X2 and X3 involving gradual cross sectional area change is as follows:

Impedance at x3 is defined by using equation (4-14) thru (4-19).

(4-22)

Since x2 = 0, therefore:

(4-23)

Or, rewriting equation (4-23) in terms of the reflection coefficient, we have:

(4-24)

Similarly, swirler impedance measured from upstream side is given by:

(4-25)

4.5.2 Upstream Swirler Impedance Measurement (Non-Reacting Case):

4.5.2.1 Validation of Upstream Transducers:

The functioning of transducers can be checked by comparing their measurement and theoretical value.

This transducer check should be mandatory before any use of the combustion instability rig. A hard wall termination is required in order to compare the theoretical reactance and measured reactance. The ideal location of a hard wall placement would be at X1 where the upstream PCB transducers are between the pressure wave source (speaker) and a hard wall. In reality, once the combustion test rig is assembled, it is not an easy task to disassemble or insert a hard wall between the upstream duct and the SAS combustor.

Although the boundary of X1 is not a hard wall, the area ratio between the upstream duct and the SAS combustor inlet is quite large; in other words, it is closer to a hard wall termination and, therefore, for checking purposes only, it is treated as a hard wall.

Part1: Using Theoretical Reactance

We verified the PCB transducers by checking theoretical reactance and measurement. Figure 4.13 shows that 0.127m (5‖) from the X1 termination is the measurement location. The procedure of measuring impedance using PCB #1, #2, and #3 at various locations is expressed in sections to follow (Section

4.5.2.2).

Figure 4.13 Upstream Calibration Check Schematic

Location of PCBs: PCB #1 = 0.5334m (21‖), PCB #2 = 0.6858m (27‖), PCB #3 = 0.7874 (31‖),

measurement location = 1.016m (40”)

Theory: where x = +0.127m (+5‖)

Figure 4.14 Comparison of the Theoretical and Measured Reactance at x=1.016m (40”)

Theoretical reactance with a cavity depth of 0.127m (5‖) created by a hard-wall and measurement of reactance at 1.016m (40‖) correlates well (Figure 4.14), and shows that upstream PCBs are functioning correctly. Note that the speaker is not capable of producing a strong acoustic signal below 200Hz approximately, and the cut on frequency of the first spinning mode (m=1) in the upstream duct is calculated to be near 1300Hz at room temperature.

Part2: Using Theoretical Transfer Function

An alternate technique to validate the function and/or calibration of PCB transducers is using the transfer function. This method does not require specifying the measurement location.

Location of PCBs: PCB #1 = 0.5334m (21‖), PCB #2 = 0.6858m (27‖), PCB #3 = 0.7874 (31‖)

Theory:

Figure 4.15 Comparison of the Theoretical and Measured Transfer Functions between PCB #2 and

Figure 4.16 Comparison of the Theoretical and Measured Transfer Functions between PCB #3 and

4.5.2.2 Impedance Measurement: Upstream Duct

Figure 4.17 Upstream Schematic (Non-Reaction Case)

The first impedance measurement location is X1U, where x = 1.143m (45‖) from the reference line x = 0.

How the Impedance of x1, z1U, Is Measured:

Obtain the transfer function, H2,1 and H3,1

(4-26)

Where

Reference location X0 is on the upstream end of the 45‖ upstream duct, therefore the PCB #1, #2, and #3 locations are:

By solving the matrix in equation (4-26), incident and reflected complex acoustic wave amplitudes A and

B are obtained.

Measuring location X1 is at 1.143 m (45‖) from the reference location X0.

Solving the above matrix gives us A1 and B1

Therefore, the impedance of the upstream side (0.154m (6.065‖) via duct) at X1 is defined by:

4.5.2.3 Impedance Measurement: Step Change at Flange:

Equation (4-18) is used to compute the impedance of x1D, which can be defined simply by applying the area ratio to impedance of upstream side x1U.

And

Impedance Plot of Z1U and Z1D obtained by using equation above:

Figure 4.18a Impedance of X1U before Step Change

Figure 4.18b Impedance of X1D after Step Change

The impedance of the flange upstream section, Z1U, and downstream section, Z1D, were plotted in Figure

4.18a and Figure 4.18b. Acoustic wave transmission experienced a sudden step area change at this section, and the relation between the upstream section and the downstream section in terms of impedance is expressed in equation (4-18). Note that area ratio was the only parameter and was directly applied to the impedance value; therefore, the change in impedance by step area changed results in increment or reduction of amplitude. The downstream section of area at X1 is much smaller than the upstream section, therefore amplitude was reduced while impedance characteristics, i.e., high/low resistance/reactance frequencies, were unchanged by step area change.

4.5.2.4 Impedance Measurement: Diffuser and Swirler

Following Equation (4-9) to (4-25), the impedance of the swirler at xSU is defined.

Figure 4.19 Schematic of Diffuser Section

How the Impedance of xSU, zSU, Is Measured:

We have already defined impedance after the step area change at x1, z1D. Now, revisiting the expression shown in equation (4-20), the impedance change from x1D to x2 can be written as follows:

Where

Impedance change from the point x2 to x3 follows the same equation:

Where

Finally, the impedance of swirler section ZSU is definied by:

Where

Impedance Plot of ZSU Obtained by Using Equation Above:

Figure 4.20 Impedance of the Swirler, ZSU, Measured from Upstream

By going through a step area change at the flange section, X1, and a few gradual area changes at the diffuser section, the acoustic wave measured by the upstream PCB transducers #1, #2, and #3 was capable of defining the impedance at the swirler measured from upstream. The measurement result is plotted in

Figure 4.20. The flat low impedance data at low frequencies up to 400Hz is significant. It means that the waves created from combustion can propagate through upstream, while reflected waves from the upstream termination at the heater could propagate back into the combustion area easily up to 400Hz.

This is the observation concluded by defining the impedance of the swirler measured from upstream; additional investigation is necessary, such as measurement from downstream, which is in the next section.

4.5.3 Downstream Swirler Impedance Measurement (Non-Reacting Case)

4.5.3.1 Validation of Downstream Transducers:

Figure 4.21 Schematic of Downstream Duct and Combustor

The location X4 is an ideal location to check the function of downstream PCB transducers. By placing the hard wall at the exhaust end, the theoretical impedance anywhere in the 1.143 m (45‖) long exhaust duct is defined with the equation Z(imag) = - cot (kL), where L is the location of interest within the duct from the hard wall termination in Figure 4.21.

Location of PCBs: PCB #4 = 2.2190 m, PCB #5 = 2.3714 m, PCB #6 = 2.4714:

- Measurement location (with PCB #4, #5, and #6) = 1.684 m

- Theory = - i cot k x, @ x = + 1.143 m

Theoretical reactance with a cavity depth of 1.143m (45‖) created by a hard wall and measurement of reactance at 1.684m correlates well ( Figure 4.22), and concludes that downstream PCBs are functioning correctly and, therefore, ready for measurements. Note that the sound wave created by the speaker is not capable of generating good acoustic signal below the 200Hz signal. The limitation of the equipment is the

91 reason for poor correlation between the theoretical and measured values seen below 200Hz in Figure

4.22.

Figure 4.22 The Comparison of Theoretical and Measured Acoustic Reactance Result Verifies the

Correct Function of the PCB Transducers #4, #5, and #6.

4.5.3.2 Impedance Measurement: Downstream Duct

First, we measured the impedance at X4 using HPCB #4, #5, and #6. The location of X4 is 1.684m.

How the Impedance of X4, Z4, Is Measured:

To obtain the transfer function H2,1 and H3,1

Where

Reference location X0 is on the upstream end of 1.143m (45‖) upstream duct, therefore the PCB #1, #2, and #3 locations are:

By solving the matrix equation, incident and reflected complex acoustic wave amplitude, A and B are obtained.

The measured location, X4 is 1.684 m from the reference location X0, in the upstream duct.

Solving the above matrix gave us A1 and B1.

Therefore, the impedance of the exhaust of combustor at X4 is defined by:

Impedance Plot of Z4 Bbtained by Using Equation Above:

Figure 4.23 The Acoustic Impedance, z4, at the Inlet, x4, of the Downstream Duct

The impedance of the section between the combustor end and the downstream duct inlet is plotted in

Figure 4.23. Obviously, there is no step change or attenuation material between the combustor and the downstream duct section; therefore the resistance is of low amplitude throughout the duct. As it was noted earlier, the broadband noise was generated by a speaker and is not capable of generating a high amplitude acoustic signal below 200Hz.

4.5.3.3 Impedance Measurement: Combustor

Figure 4.24 Schematic of Combustor Section

The combustor section experiences diverging and converging paths (Figure 4.24), therefore, the impedance per section is obtained by marching through the area change per cooling liner. The end of each cooling liner section is chosen to be the location of the step point. Point A is the swirler cup region, and point B is the end of the first cooling liner, and so on. From the end of flange (X4) to the swirler (XSD), there will be 10 steps of gradually changing cross-sectional areas.

The inner liner in the combustor has a gradual cross-sectional area change as well as a step change.

Hence, the impedance measured at X4 needs to go through a step change as well as a gradual area change.

Cooling Liner “Gradual Area Change and Step Change”

Figure 4.25 Schematic of Combustor Cooling Liner Section

How the Impedance of xSD, zSD, Is Measured:

From the x4, the geometry of the combustor expands and then diverges. The change in the cross sectional area is the repeat of gradual change, and then step change, as shown in Figure 4.16.

Where

Note*Delta L contains a NEGATIVE sign, because we were marching from downstream to upstream.

The impedance change from point xI to xH follows the same equation:

Note* HU stands for upstream end of the line ―H‖, and HD stands for downstream end of the line ―H‖

Where

Step area change, gradual area change … and so on from H to A.

Finally the impedance of swirler section, zSD is definied by:

Where

Impedance Plot of ZSD (swirler Impedance measured from downstream) Obtained by Using the

Equation Above:

Figure 4.26 Impedance of the Swirler, ZSD, Measured from Downstream

The acoustic waves measured by the downstream PCB transducers #4, #5, and #6 let us obtain the impedance of the swirler measured from downstream (Figure 4.26), by going through series of step and gradual area changes at the combustion section (Figure 4.25). The result of swirler impedance measured from downstream shows mostly flat low impedance across the frequency up to above 1000Hz.

4.5.4 Swirler Impedance

4.5.4.1 Swirler Impedance by Acoustic Pressure Wave

Finally, we have the impedance of the swirler measured from upstream and downstream. Change in the impedance (Delta Z) should give us approximate swirler impedance itself.

Therefore

Figure 4.27 Impedance of the Swirler, ΔZ, Full Frequency Scale

Impedance of swirler, ΔZ, is plotted in Figure 4.27. It shows the flat low impedance up to near 400Hz.

Increased impedance is visible from 400Hz to near 500Hz and then the impedance plot shapes into a complicated shape. Our interests were located in a lower frequency, since combustion instability occurs in low frequency oscillations; therefore we can ignore the impedance characteristics of the high frequency portion.

Figure 4.28 Impedance of the Swirler, ΔZ, Selected Frequency Range

As it was mentioned earlier, our interests were concentrated in lower frequency ranges. This is because the microphone was stationed near the combustion; measuring radiating sound waves from the combustor showed peaks in low frequency ranges, and chemiluminescene analysis also indicated low frequency heat release oscillation during combustion instability. Figure 4.28 is a plot of the swirler impedance measured by PCB transducers, i.e., acoustic pressure waves. The flat line of the real part of the impedance shows the low or zero resistance in the swirler throughout the low frequency range, letting low frequency oscillation caused by combustion propagate through the swirler to upstream. This result also verified the reason why our combustor has combustion instability with the frequency below 400Hz.

100

4.5.4.2 Swirler Impedance by Pressure Drop

The pressure drop across the swirler as a parameter is a common term to verify the experimental condition; at the same time, pressure drop is simply the resistance of the swirler itself. We can conduct approximate swirler impedance by the only information given by pressure drop across the swirler and its area.

Swirler impedance is given by:

(4-27)

Where

The resistance caused by the swirler, or a porous septum/sheet is ―R‖ in the above equation, which is a normalized resistance, and ―m‖ is a mass reactance. Because of high temperature during combustion, the speed of sound, c, is significantly high; at the same time, mass-reactance is a typically low value, therefore the imaginary term of equation (4-27) becomes negligible.

Figure 4.29 Pressure Drop Across Swirler vs. Mass Flow Rate

101

Figure 4.30 Resistance of the Swirler Obtained from Pressure Drop

102

4.5.5 Pressure Wave Measurement in the Combustion Area

Four PCB pressure transducers were placed on the sidewall of the combustor (Figure 4.31). The purpose of these transducers is to provide pressure oscillation in detail during combustion, and locations are selected carefully.

Figure 4.31 Schematic of Downstream Duct and Combustor with Four PCB Transducers

Figure 4.32 Schematic of PCB Transducers on the Combustor Wall

103

Selected locations of PCB transducers on the combustor wall are shown in Figure 4.32. Locations are kept at equal distance from the centerline of flow path while they are kept as close as possible to the cooling liner walls in each direction. PCB transducer locations in longitudinal direction are determined by observing chemiluminescene images obtained by high speed video. For example, if circulation exists during combustion, PCB #1c and PCB #2c will have a large phase difference; in other words, the configuration is capable of measuring the transverse mode (normal flow direction) wave propagation if it exists during combustion.

104

4.5.5.1 Plane Wave Measurement Using Top Half of PCB Transducers on the Combustor

Wall

Compare Measurement vs. Theory at X1C from PCB1c & 3c

Figure 4.33 Top Half of PCB Transducer on the Combustor Wall

Impedance at X1C was obtained by using PCB #1c and #3c transducers (circled in the schematic of Figure

4.33) because we were interested in the propagating wave in the longitudinal direction, and the cut on transverse mode frequency in the combustor is above our frequency of interest. Since the cross sectional areas of X1C and X3C are not equal, gradually changing cross sectional area methods needed to be introduced during the computation of A1C and B1C. Referring to equations shown in the earlier section

―Gradual Area Change,‖ we have:

Where

105

And we have the measured impedance value at X1C:

Next, we needed to obtain the theoretical impedance. By placing the hard wall at the end of the exhaust duct with the length of 1.143m, we were able to obtain theoretical impedance value at X4.

The acoustic wave experience diverging path toward x2C, then converging path to x1C from x4, therefore for accuracy, two step area changes should be considered.

Where

And we have the theoretical impedance value at X2C:

Continue the same procedure,

Where

and we have the theoretical impedance value at X1C:

106

Figure 4.34 Theory vs. Measurement of Reactance Measured at X1C Using Top Half Section

Measurement data was compared with the theoretical value and plotted in Figure 4.34. To increase the accuracy of the sound wave generated, a tonal signal was excited instead of broadband excitation. The correlation between the theory and measurement was not excellent. The difference (error) could be in the misunderstanding of the theory, caused by an approximation of geometry, or other factors that are considered as negligible during measurement. Further investigation should be conducted on this task.

107

4.5.5.2 Plane Wave Measurement Using Bottom Half of PCB Transducers on the

Combustor Wall

Compare Measurement vs. Theory at X1C from PCB2c & 4c

Figure 4.35 Bottom Half of PCB Transducer on the Combustor Wall

The impedance at X1C can be also obtained by using the PCB #2c and #4c transducers (circled in the schematic of Figure 4.35) because we were interested in the propagating wave in the longitudinal direction, and the cut on transverse mode frequency in the combustor was above our frequency interests.

The procedure for obtaining the impedance at X1C is exactly same as the procedure with PCB#1c and #3c except the transfer function.

Where

And we have the measured impedance value at X1C:

108

Next, we needed to obtain the theoretical impedance. By placing the hard wall at the end of the exhaust duct with the length of 1.143m, we are able to obtain theoretical impedance value at X4.

The acoustic wave experienced a diverging path towards x2C, then a converging path to x1C from x4; therefore for accuracy, two step area changes should be considered.

Where

And we have the theoretical impedance value at X2C:

Continue the same procedure

Where

109

and we have the theoretical impedance value at X1C:

Figure 4.36 Theory vs. Measurement of Reactance Measured at X1C Using Bottom Half Section

Measurement data was compared with theoretical value and plotted in Figure 4.36. As in the measurement conducted with the top half of transducers, measurement with the bottom half of transducers did not show good agreement with the theoretical value. As it was suggested earlier, further investigation should be conducted on the issue of disagreement of theoretical and measurement value.

110

4.5.5.3 Plane Waves and First Transverse Mode Measurement in the Combustor

Figure4.37 Schematic of Combustion and Propagating Waves

To understand more detail in the combustion area, four PCB transducers were placed as shown in the figure above, Figure 4.37. The transducers were set up so that they were capable of measuring two acoustic modes that may have propagated in both directions in the combustion area. The diagram above shows the two opposite pressure transducer locations along the centerline, X1C and X2C. Each pair is equally spaced from the centerline.

The equation of acoustic pressures measured by these transducers mounted on the combustor walls are:

(4-28)

(4-29)

(4-30)

(4-31)

111

Transforming equation set (4-28) to (4-31) to matrix equation:

Plane Wave:

(4-32)

First Transverse Mode:

(4-33)

Equations (4-32) and (4-33) can be written in terms of the transfer function data, as follows:

(4-34)

)

(4-35)

Figure4.38 Theory vs. Measurement of Reactance Measured at X1C Using all Four Transducers

112

Measurement conducted using all four PCB transducers was compared with theoretical value and plotted in Figure 4.38. As with the other measurements conducted with the top half and bottom half of transducers, agreement between measurement and theoretical value is poor. It is highly recommended that additional investigation be conducted on the issue of disagreement of theoretical and measurement value.

Since the sound source was generated upstream of the swirler, i.e., outside of the combustor, higher frequencies that could be the first transverse mode were not propagated through the swirler. Equation (26) is the set for the first transverse mode measurement, but it is meaningless in this measurement configuration.

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4.5.6 Upstream and Downstream Boundary Condition:

Figure 4.39 Measurement of Upstream Boundary Condition, Air Heater

In order to measure the upstream boundary condition using PCB transducers on the upstream duct, the speaker must be placed on the downstream side of the array of PCB transducers, as shown in Figure 4.39.

This is because the sound wave source (speaker) itself creates its own impedance, which is unknown. The speaker mount hole on the upstream section of the duct is plugged, which gives a smooth surface on the inner duct giving us ideal conditions under which to measure the impedance of the heater.

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4.5.6.1 Air Heater Impedance Calculation:

The air heater contained two different cross sectional areas. The procedure of impedance calculation over the step area changes were expressed earlier in equations (4.13) to (4.18).

Step Area Change at Air Heater Flange:

Two necessary conditions that need to be satisfied for step change were:

1) Continuity of pressure

2) Continuity of volume displacement (conservation of mass)

Therefore, corresponding equations for the above conditions at boundary x0 gave us

―D‖ represents the downstream side, ―U‖ represents the upstream side.

Where

Using the relation expressed in the above equation sets, impedance of the upstream side at x0 is obtained.

Procedure of Air Heater Impedance Calculation from the X0:

115

How the Impedance of x0, z0D, Is Measured:

Obtain the transfer function, H2,1 and H3,1

Where

Reference location X0 is on the upstream end of the 45‖ upstream duct, therefore the PCB #1, #2, and #3 locations are:

By solving the matrix above, incident and reflected complex acoustic wave amplitude A and B are obtained.

Measuring location X0 is the reference point, therefore:

116

Solving above matrix gives us A0 and B0

Therefore, the impedance of the upstream side (6.065‖ via duct) at X0 is defined by:

And step area change:

Where

Shifting the location to the end of the wide flange area section:

Where

And step area change:

Where

Shifting the location to the air heater end:

117

Where

Flow Control Valve Opening: 0 Turn

Figure 4.40 Measured Impedance at X0D, the Upstream Boundary Condition

Impedance at X0D is measured and plotted in Figure 4.40. The flow control valve opening is set to zero during measurement.

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4.5.6.2 Upstream Mass Flow Control Valve

Figure 4.41 Schematic: Mass flow Control Valve Location

The opening of the upstream mass flow control valve could change the air heater impedance. The valve rotates eight and half turns; therefore, nine sets of impedance data for the air heater impedance were observed. A comparison of valve opening versus impedance value is shown in Figure 4.42 and Figure

4.43.

Figure 4.42Resistance of Heater vs. Mass Flow Control Valve Openings

119

Figure 4.43 Reactance of Heater vs. Mass Flow Control Valve Openings

The resistance and reactance portions of the impedance of the air heater with various valve opening ranges are plotted in Figures 4.42 and 4.43, respectively. Results showed that the effects of the mass flow control valve opening levels were negligible at higher frequency ranges of impedance, while some differences were seen in lower frequencies.

120

Is the Heater Acting Like a Hard Wall?

No significant change in the impedance characteristic of the heater end surface by changing the upstream valve opening amounts led to the possibility that the heater was acting similarly to the hard wall condition. Therefore, comparing the theory (hard wall) with the measurement shows how much the heater was acting similar to a hard wall.

Figure 4.44 Schematic of Heater Section

If the heater surface, XH is the hard wall (Figure 4.44), then we have:

A hard wall was placed at the inlet of the diverging tube adapter, as shown in Figure 4.45. The purpose of this test was to determine the effect of the upstream beyond the diverging tube adapter that was connected to the air heater. If the impedance value of the air heater did not greatly change with or without a hard- wall being placed at the diverging adapter inlet section, then the significance of the acoustical wave path condition beyond the air heater, such as the air flow valve and flow meter, can be concluded as small.

121

Figure 4.45 A Hard Wall Is Placed at the Upstream End Section of the Air Heater

Figure 4.46 Reactance of Section X-1

122

Figure 4.46 is a plot of reactance at termination X-1 with various air flow valve openings, a case of hard wall placed, and theory of the air heater as a hard wall. There were some differences between the measurements and the theoretical line; however, the measurement follows the trend of the hard wall.

123

4.5.6.3 Exhaust Duct (X5) Radiation Impedance Calculation

The downstream boundary section of the combustion rig was the end of the exhaust duct (downstream duct), which was open to the atmosphere (Figure 4.47). In this case, the impedance of the duct section open to the atmosphere was not simply zero, but has impedance, the so-called radiation impedance.

Figure 4.47 Exhaust Duct End

A great deal of experimental work [26] has yielded an empirical correlation for the radiation impedance of open flanged or unflanged circular pipes.

This correlation of open unflanged circular pipe is as follows:

(4-36)

Where

The cross sectional area of the rectangular exhaust duct is 0.00585 m2, and the equivalent area of circular pipe has a radius, r, of 0.043152 m. By introducing this radius into the equation above, the impedance value of unflanged circular pipe of radiating impedance can be calculated and plotted in Figure 4.48.

124

This correlation of open flanged circular pipe is as follows:

(4-37)

Where

A rectangular flange area at the end of the exhaust duct is measured to be 0.01579 m2, but the flange area is irrelevant for the theoretical flanged circular pipe radiating impedance calculation. By introducing this radius into the equation above, the measured and theoretical value of flanged circular pipe of radiating impedance can be calculated and plotted in Figure 4.49.

Both plots in Figure 4.48 and 4.49 indicated that small error in measurement and theoretical value. This is likely caused by a difference of shapes between the circular pipe and the rectangular duct, or an error in ambient temperature value measured.

Figure 4.48 Measured and Theoretical (Unflanged Circular Duct) Radiating Impedance

125

Figure 4.49 Measured and Theoretical (Unflanged Circular Duct) Radiating Impedance

126

4.5.7 Impedance Calculation with Moving Fluid

4.5.7.1 Effect of Fluid Flow Within Experimental Range

So far, we have made measurements while placing a hard wall on the upstream and downstream so that the upstream and downstream boundary conditions were known.

Figure 4.50 Schematic of Entire Combustor Rig

Area 1 and velocity 1 is location at the end of the heater, and area 2 and velocity 2 correspond to the exhaust duct, from X4 to X4.

Pound/s kg/s Rho(kg/m^3) Area 1(m^2) Velocity 1(m/s) Area 2(m^2) Velocity (m/s) 0.05 0.02268 1.174 0.018241 1.059 0.00585 3.302 0.08 0.03629 1.174 0.018241 1.694 0.00585 5.284 0.12 0.05443 1.174 0.018241 2.542 0.00585 7.925 0.16 0.07257 1.174 0.018241 3.389 0.00585 10.567 0.18 0.08165 1.174 0.018241 3.813 0.00585 11.888

Throughout the rig, the mass flow rate was constant, but the Mach number changed locally by its cross sectional area and fluid temperature.

Up to this point, we kept our wave number, k, as a constant anywhere in the rig as well as both directions of complex acoustic wave amplitude, A and B, but that assumption does not apply with moving fluid.

Replacement for the constant wave number, k, is the propagation constant, κ (kappa):

127

For forwarding wave (+), A, we have:

(4-38)

For reflected wave (-), B, we have:

(4-39)

Therefore, acoustic pressure and particle velocity equations were rewritten as follows:

(4-40)

(4-41)

Figure 4.51 Resistance at X4 vs. Various Mass Flow Rates

128

Figure 4.52 Reactance at X4 vs. Various Mass Flow Rates

Figure 4.53 Resistance at X5 vs. Various Mass Flow Rates

129

Figure 4.54 Reactance at X5 vs. Various Mass Flow Rates

An assumption was made that the effect of moving fluid in our experiment range did not affect the impedance values in various sections because the Mach number was low, which was near negligible range. To observe the effect of mass flow rate change in impedance, the location of X4 and X5 were chosen for demonstration purposes. The results shown in Figure 4.51 to Figure 4.54 indicate that there were no significant changes in impedance values from no fluid flow to the maximum of 0.18 pounds per second (pps).

130

4.5.7.2 Effect of Air Temperature on Impedance:

Figure 4.55 Schematic of the Combustor Rig with Thermocouples Installed

Since the speaker is placed on the upstream end of the PCB transducers, impedance of the heater, XH, or

X-1, and X0, cannot be measured as configured in the figure above.

Measured locations are: X1U, XSU, XSD, X4, and X5

FlowRate (pps) Temperature (F)

0.09 200F 300F 400F 500F 600F

Note* For heater impedance measurement, the speaker is placed on the downstream side of the 6‖ duct.

Impedance at various sections of the combustion rig with different air temperatures were measured and plotted in Figure 4.57 to 4.68. Unlike changes in flow rate, which did not affect impedance, changes in air temperature did affect the impedance in both amplitude and shift in peak frequencies. First, this was caused by the change in wave number, k.

For demonstration purposes, the impedance at XSD, swirler impedance measured from downstream, was converted from a function of frequency to wave number to eliminate variations caused by temperature.

Resistance and reactance of swirler impedance measured from downstream in terms of wave number were

131 plotted in Figure 4.69 and 4.70, respectively. It became much more uniform than the plot with frequency as a function. This way, the normalized uniform value of impedance throughout the combustion rig can be obtained and used for simulation purposes.

132

Figure 4.56 Resistance of Heater vs. Various Air Temperatures

Figure 4.57 Reactance of Heater vs. Various Air Temperatures

133

Figure 4.58 Resistance at X1U vs. Various Air Temperatures

Figure 4.59 Reactance at X1U vs. Various Air Temperatures

134

Figure 4.60 Resistance at XSU vs. Various Air Temperatures

Figure 4.61 Reactance at XSU vs. Various Air Temperatures

135

Figure 4.62 Resistance at XSD vs. Various Air Temperatures

Figure 4.63 Reactance at XSD vs. Various Air Temperatures

136

Figure 4.64 Resistance at X4 vs. Various Air Temperatures

Figure 4.65 Reactance at X4 vs. Various Air Temperatures

137

Figure 4.66 Resistance at X5 vs. Various Air Temperatures

Figure 4.67 Reactance at X5 vs. Various Air Temperatures

138

Figure 4.68 Resistance as a Function of Wave Number at XSD vs. Various Air Temperatures

Figure 4.69 Reactance as a Function of Wave Number at XSD vs. Various Air Temperatures

139

Chapter 5: Result and Discussion – Rayleigh’s Criterion (p’ & q’)

Rayleigh Index:

A well known criterion established by Lord Rayleigh is that combustion instability occurs when heat is released while the pressure wave amplitude is at its maximum. In other words, instability is amplified if heat release and pressure oscillations are in phase.

(5.1)

If G(x) >0, then it is unstable combustion: amplitude of pressure wave increases

If G(x) <0, then it is stable combustion: amplitude of pressure wave decays

In Chapter 3, two cases were introduced:

Case 2: High fuel-to-air ratio (f/a=0.0343) tested on the newly designed rig

Case 3: Low fuel-to-air ratio (f/a=0.008) tested on the newly designed rig

Pressure Wave Oscillation (p’):

PCB pressure transducers mounted on the combustion side wall were capable of providing pressure wave oscillation information, p‘. In Chapter 4, Figure 4.7 and Figure 4.8 showed that the spectrum of four individual transducers on the combustor obtained similar spectrums, indicating that no circulation or significant entropy wave was generated during combustion, i.e., only the plane wave existed during combustion. Our interests were not local points of oscillation but general characteristics of the entire combustion, therefore pressure wave oscillation values were the average of four PCB transducer waveforms.

Heat Release Oscillation (q’):

Chemiluminescene analysis provided heat release shape, location and rate during combustion by its color intensity. As it was explained in Chapter 3, POD is an excellent method to obtain the most dominant

140 motion within a highly complex field, in other words, filtering out noise components but giving the most general information of combustion heat release shape, location and rate. The first three most dominant

POD mode shapes and corresponding time-domain waveforms of modes for Cases 2 and 3 were shown in

Chapter 3, Figure 3.21 and Figure 3.22 for Case 2; and Figure 3.25 and Figure 3.26 for Case 3. For both cases, mode #1 had the highest energy, therefore heat release oscillation q‘, was obtained from the waveform that corresponded to mode #1.

Phase Relation:

The phase relation between pressure and heat oscillation was defined by using a complex transfer function defined as:

(5.2)

The real component of equation (5.2) gave a relation of magnitude (=1 being exactly the same amplitude), and the imaginary component is the phase relation (=0 being in phase), in the function of frequency.

Results of Rayleigh Criterion Identification:

Although distinct peak amplitude at a particular frequency existed in Case 2, the phase angle between pressure oscillation, p‘, and heat release, q‘, did not match, i.e., were out of phase, as shown in Figure 5.1.

Note that the peak amplitude shown in Figure 5.1 and 5.2 are calculated by p‘(f) * q‘(f) and then normalized. Throughout the investigation of Case 2 using chemiluminescene, as well as pressure waves, in Chapters 3 and 4, there were no significant hints of combustion instability occurring during Case 2.

Opposed to the Case 2 result, there were various strong hints of combustion instability from chemiluminescene and pressure wave analysis from the Case 3 experiment. Peak amplitude frequency

(normalized as peak =3.14) and phase relation of p‘ and q‘ in the Case 3 experiment were plotted in

Figure 5.2., which shows that the phase angle between p‘ and q‘ are in phase from 50Hz to 175Hz, while

141 the peak frequency sits in middle. This result follows the Rayleigh Criterion; therefore, Figure 5.2 is proof that combustion instability was occurring during the Case 3 experiment.

Figure 5.1 p’ and q’ Phase Difference and Normalized Peak Frequency Amplitude for Case 2

Figure 5.2 p’ and q’ Phase Difference and Normalized Peak Frequency Amplitude for Case 3

142

Chapter 6: Modeling of Acoustic Resonance Modes

6.1 Introduction

The method of predicting combustion instability frequency by modeling the experimental combustor rig is introduced in this chapter. The condition that is causing combustion instability, Rayleigh‘s criterion, is explained in Chapter 5. The potential combustion instability case, Case 3, was subject to further investigation to validate Rayleigh‘s criterion, i.e., the occurrence of combustion instability. The result provided a confirmation of instability, expressed in Figure 5.2; however, it suggested that the occurrence of instability was only within the range from 50Hz to 175Hz. During the Case 3 combustion experiment, various pressure transducers and a high speed camera (chemiluminescene imaging) captured the oscillation with a fundamental frequency of a near 106Hz single peak along with its harmonics, 214,

322Hz, and so on, as shown in Figure 3.24, Figure 3.25, and Figure 4.8 The reason the frequency was limited to 106Hz was because it is governed by not only Rayleigh‘s criterion but also the geometry of the combustor rig.

Successful modeling of combustion acoustic dynamics should allow for the prediction of the frequency and the amplitude of the combustion instability. The mathematics utilized in this modeling is a modified version of a constant cross sectional duct containing an acoustic source, as shown in Appendix II or [28].

The modeling requires the information of the upstream/downstream boundary impedance conditions, swirler impedance, and the geometry of the combustor rig, and also the temperature and velocity of the air moving across the combustor rig. Fluid temperature plays a significant role in determining the speed of sound, while the contribution of fluid velocity is small because the Mach number is quite low in the particular case being discussed. The upstream and downstream boundary conditions of the combustor rig, as well as the swirler impedance, were defined in Chapter 4. Figure 6.1 illustrates the boundary locations of the combustion rig. In an earlier chapter, the simplified theoretical impedance value of the upstream boundary condition at the air heater and downstream boundary at the exhaust area were compared and it

143 was confirmed that the result is not exact but a similar value when comparing theory and measurement; therefore, they can be used as approximated boundary conditions. In this chapter, both modeling with simplified theoretical boundary conditions and with actual measured boundary conditions were conducted to show how close the modeling can be in predicting combustion instability frequency and location with approximated boundary conditions.

Figure 6.1 Schematic of the Rig within Boundaries and Acoustic Source Location

6.2 Modeling of Acoustic Resonance Criterion/Theory

The criterion of acoustic resonance described below is based on theroretical analysis by Dr. Asif Syed.

This analysis is presented in his notes contained in Appendix II or [28]. In Chapter 4, a derivation of acoustic impedance from downstream and upstream toward the swirler to measure the swirler impedance is expressed. Imagine that combustion (acoustic velocity source) is located at a discrete location as shown in Figure 6.2 as the first approximation. Instead of shifting the measurement line toward the swirler ends as performed in Chapter 4, let the measurement location shift toward the acoustic source, i.e., the combustion source line from the upstream boundary end and the downstream boundary end.

144

Figure 6.2 Illustration of Discrete Combustion Location

Across the discrete combustion source line, xcombustion, boundary conditions to be satisfied are:

1) Continuity of acoustic pressure

2) Conservation of mass

Let the incident and reflected complex acoustic wave amplitudes between the upstream boundary and the discrete combustion location be AU and BU, respectively, while the wave amplitudes between the downstream boundary and the combustion location be AD and BD. Acoustic pressure, velocity, and impedance in terms of incident and reflected complex acoustic wave amplitudes are expressed in equation

(4.1) and equation (4.2) in Chapter 4.

Let ∆x1 and ∆x2 be the distances between the upstream boundary to the source plane, and the downstream boundary to the combustion source line respectively.

From the continuity of acoustic pressure, relation can be written as:

(6-1)

From the conservation of mass:

(6-2)

Divide equation (x2) by acoustic pressure wave between the source and downstream boundary gives:

(6-3)

145

Using relation in equation (x1), equation (x3) can be rewritten as:

(6-4)

Equation (x4) can be rewritten as:

(6-5)

and

(6-6)

Where is acoustic pressure at the acoustic source (combustion) equation (x6) states that the right hand

side denominator, , becomes close to zero, then right hand side, , pressure term becomes maximum. Therefore, resonance or combustion instability occurs when:

(6-7)

The acoustic impedance values at the upstream and downstream boundaries are measured by PCB pressure transducers simultaneously during an experiment. Therefore, the actual boundary conditions can be utilized in the modeling.

In Chapter 4, the impedance change due to the swirler was determined by two different methods. One was by computing ∆ZS from the acoustic measurement in the combustor rig, and the other method was based on the DC flow method. This impedance was obtained from the pressure drop across the swirler.

Impedance results by those two methods were shown in Figure 4.28 and Figure 4.30 respectively. (See

Chapter 4 for details of those two measuring techniques.) Both methods concluded that the impedance change due to the swirler was nearly zero over the frequency range of interest. Therefore, the impedance of the swirler was treated as negligible, i.e, zero, as the first approximation for the modeling. Temperature values used in the modeling were the measured values during the experiment and separated into four

146 segments. Each type-K thermocouple was responsible for its temperature segment, as shown in Figure

6.3. It is preferable to introduce the detailed temperature profile throughout the combustor rig; however, temperature was constant within its own segment as the first approximation, and for simplicity.

Figure 6.3 Temperature Segments in the Combustor Rig

The first estimate of the combustion acoustic source location was set at 0.02m from the swirler cup end.

The reason the combustion location (acoustic source), xcombustion, was set to be very close to the swirler is because the flame front of the lean combustion is typically very close to the swirler.

6.3 Acoustic Resonance Modeling Result

The approximate combustion instability location and frequency were modeled with inputs of temperature values in four segments, upstream and downstream boundary conditions, impedance of the swirler, air mass flow rate, and the geometry of the combustor rig. Note that the impedance of the swirler was found to be low; therefore it was treated as negligible for the modeling. Recall that the subject case, Case 3, had a combustion instability frequency of 106Hz, identified by both pressure transducers and high speed camera (chemiluminescene) data. Plotting equation (6-7) in the frequency domain, the combustion instability frequency can be identified where the difference of impedance across the combustion source line reaches near zero. Figure 6.4 shows the result of the combustion instability prediction model for Case

3 using equation (6-7). The value was calculated by using actual boundary condition values measured during the Case 3 combustion test, since those values were simultaneously measured by PCB pressure

147 transducers. As it is seen clearly in Figure 6.4, the value is lowest at 110Hz; in other words, the modeling result predicts that the combustion instability occurs at 110Hz for the Case 3 condition. The actual combustion instability frequency was 106Hz. Therefore the predicted frequency of the combustion instability is considered to be close to the measured value even though several simplifying assumptions were made in the theory for the modeling. In addition, note that the frequency resolution in the modeling is 5Hz. This is because of the aim of utilizing a boundary impedance condition data set that has higher accuracy by increasing the FFT average number. The result of gaining accuracy was the trade off of the frequency resolution. Figure 6.5 is the result of the combustion instability prediction model using simplified theoretical boundary conditions (hard wall termination at upstream boundary of the air heater end surface, and empirical flanged radiating impedance [26] value at downstream boundary of the exhaust tip area). The use of simplified ―theoretical‖ boundary conditions allowed the modeling procedure to move one step toward the first approximation without costly experiments using compressed air, electricity for heat and fuel to obtain boundary impedance conditions for the modeling of acoustic resonance, or prediction of combustion instability frequency. The result of the model of Case 3 with simplified theoretical boundary condition values (Figure 6.5), shows the predicted combustion instability frequency to be slightly higher than the measured value; however, as described earlier, the modeling result was obtained by a number of approximations including the combustion source location, constant temperature profile, and simplified geometry by ignoring combustor cooling flow liners, etc. If the exact modeling, i.e., defining the exact instability frequency and location, is required to be defined by the modeling, then including the precise temperature profile, exact geometry, etc., is recommended. As the first approximation, the modeling showed that the combustion instability frequency was significantly close to what was measured by pressure transducers and high speed video data (chemiluminescne). Therefore, it can be concluded that the modeling for predicting combustion instability frequency and location was successful.

148

Figure 6.4 Instability Prediction from Model Using Measured Boundary Condition

Figure 6.5 Instability Prediction from Model Using Simplified Theoretical Boundary Conditions

149

Chapter 7: Suggested Future Work

7.1: Fuel, Swirler, and Test Conditions

Although a wide range of measurement techniques, devices, and analysis methods were introduced to understand combustion instability in the combustion rig, propane is the only fuel used in the new (long) combustion rig designed for studying combustion instability. The combustor is capable of firing on not only propane but gaseous fuel or liquid fuel, such as methane, Jet-A, etc., as long as the swirler is interchanged to the correct type. Similarly, the swirler type that was introduced in this thesis was limited to one particular model, although there exist several different types of swirlers that are designed to fit in the combustor rig, and they are relatively quick and easy to exchange. Additionally, the conditions for the tests that were conducted for this thesis were limited to only two cases with fixed air heater temperature and air mass flow rate. Continuing the studies introduced in the earlier chapters with different fuels types, swirlers types, air heater temperature, and mass flow rate/pressure drops across swirlers will let researchers obtain additional data for comparison and further analysis of combustion instability behavior.

7.2: Changing Impedance of the Pressure Wave Path

The reason constant cross sectional area ducts are placed on the upstream and downstream sections of the single annular sector combustor is to simplify acoustic measurements. For instance, when the sound speaker is not being used in the upstream duct, its mounting hole is filled with a plug with a curved surface so that it does not introduce any cavities on the path of a pressure wave. The schematic in Figure

7.1 shows the proposed location and mechanism for changing the impedance of the pressure wave path.

Because of the large area of the hole and a flange section already exist to mount and operate a speaker, introducing a piston mechanism would be relatively easy. Also, a similar piston mechanism is suggested to be introduced on the exhaust duct section with the same reasoning; changing the impedance of the pressure wave path. Not only is the upstream duct section high in temperature, but also the exhaust area

150 will be extremely high in temperature; therefore the piston mechanism is suggested to be a threaded rod style so that there is nothing but steel used in the mechanism.

Figure 7.1 Schematic of the Piston Mechanism to Change Impedance of the Pressure Wave Path

Changing the impedance of the path of acoustic waves within boundaries is believed to affect the combustion instability behavior. It is clear that complex acoustic wave amplitudes change across the cavity, shown as the location in Figure 7.1 and bounded by xPD and xPU, are no longer equal, hence, altering the interaction of the acoustic source and propagating waves‘ behavior. Changing the cavity depth and location will let the researcher gain additional information on combustion instability.

7.3: Feedback Loop

Broadband or tonal signal was generated by the SRS Model DS335 Function Generator and outputted through a speaker that is mounted on the upstream duct to obtain the impedance of the swirler and boundary conditions. Although the SRS DS335 Function Generator can be controlled as a standalone unit, it is controlled by LabView program via RS232 (see Chapter 2, Figure 2.6 for the experiment wiring diagram with a signal generator). This set up is to introduce active control functionality during combustion instabilities. For example, if the combustion instability is occurring with a single peak frequency of 106Hz, then it may be of the researcher‘s interest to output 106Hz through a speaker. The

151 one computer is responsible for acquiring/analyzing occurrences through pressure transducers as well as generating signals (to speaker); therefore it is possible to precisely output a tonal signal of 106Hz in or out of phase, or anywhere between, to observe any behavior change in combustion instability. Also, if the electrically controlled fuel flow valve is introduced, then the same controlling method as a tonal signal generation can be applied to make the fuel flow pulsate. A diagram in Figure 7.2 shows feedback control of sound and fuel flow in the combustion rig. If the swirler is a dual orifice swirler, then water injection to control combustion instability is possible. The water flow for the injection can be actively controlled by using the feedback loop control. The combustion rig introduced in this thesis is already configured with the feedback loop in mind; hence, the experiments and studies of open loop feedback control to eliminate or study combustion instability behavior are possible without significant effort for setting up the rig.

152

153

Conclusion

Conclusions are made in three different sections and summarized as follows:

Chemiluminescenes of high speed video images of combustion were subjected for analysis by using methods of Proper Orthogonal Decomposition (POD) and Fast Fourier Transform (FFT). The results from

FFT on chemiluminescene images identified heat release peak locations and phase relations within a two- dimensional domain of combustion, while POD analysis provided the highest to lowest energy mode shapes of combustion in a two-dimensional form and its mode-corresponding time-domain waveforms.

POD acts as a spatial filter by identifying dominant mode shapes and its time domain oscillation in such a complicated field; in other words, the dominant heat release rate (q‘) during combustion instability can be obtained from such complex and noisy data. The findings of q‘ by the POD method contributed a significant part for identifying and confirming Rayleigh‘s criterion.

Arrays of pressure transducers that are mounted throughout the combustor rig captured pressure waves within the boundaries as well as the pressure oscillation rate (p‘) from the combustor wall. Other than local pressure oscillation measurements, the techniques and configurations described in this thesis led to the establishment of a rig system that defines the swirler impedance, as well as the impedances at air heater and open-end of the exhaust section as boundary conditions. Also, changes in impedance by temperature and air flow rate were examined. It is important to understand the behavior of pressure waves at the boundaries in order to identify acoustical resonances of the combustion rig as one of the combustion instability conditions to be met.

Phase relation of believed-to-be combustion instability and non-instability cases were examined with measured heat release rate (q‘) and pressure oscillations (p‘) using techniques introduced in this thesis. As the result, Rayleigh‘s criterion was confirmed; the combustion instability occurs when the heat release

154 rate and pressure oscillation rate are coupled. Another condition to be met for combustion instability is the acoustic resonance of the combustion rig. Using the data of boundary conditions and temperature obtained during the combustion experiments and fixed geometrical information of the combustion rig, the acoustic resonance modes were completed using one-dimensional wave theory. The result stated that it is capable of identifying acoustic resonance frequency that corresponds to the fundamental frequency measured or observed during actual combustion instability.

155

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160

Appendix I

Gradual Area Change Measurement Calculation using Wave Tube

Incident and reflected complex acoustic wave amplitudes are determined by using two transducers mounted on the gradual area change conduit.

Where

The Specific impedance at x1 measured is:

And Measured Acoustic Impedance (pressure/volume velocity) is given by:

The theoretical Specific impedance at x1 is:

And Theoretical Acoustic Impedance (pressure/volume velocity) is given by:

161

See Two Plots Below

Because the measured impedance and theoretical values are divided by the same area value, the only change appears in the Acoustic Impedance plot from specific impedance is the change in amplitudes.

162

Arbitrarily Location (M):

And

Specific impedance:

Where and

Acoustic impedance:

The Theoretical Specific impedance at x1 is:

And Theoretical Acoustic Impedance (pressure/volume velocity) is given by:

See Two Plots Below

163

164

Arbitrarily Location (N):

And

Specific impedance:

Where and

Acoustic impedance:

The Theoretical Specific impedance at x1 is:

And Theoretical Acoustic Impedance (pressure/volume velocity) is given by:

See Two Plots Below

165

166

Appendix II

THEORETICAL NOTES FOR ONE-DIMENSIONAL (1-D) RESONANCE MODES IN A DUCT OF UNIFORM CROSS SECTION, CONTAINING AN ACOUSTIC SOURCE.

Written By Dr.Asif Syed

R (f) 1 R2(f) Z1(f) us(f) Z (f) 2 A C

B D

L1 L2 x

Figure 1. The diagram of one-dimensional wave propagation in a tube containing an acoustic source (at x = L1) and with known termination impedance, Z1 and Z2, at the two terminations.

The diagram in Figure 1 illustrates a duct of constant cross section in which an acoustic velocity source, us(f), is located at location x=L1.

The duct contains one-dimensional acoustic wave propagation, which is driven by the acoustic velocity source. The duct terminations at the two ends are assumed to be reflective. The reflection coefficients, R1(f) and R2(f) at the two ends, are assumed to be known. The objective of this analysis is to model the acoustic resonance modes in this duct.

Let us consider the acoustic field in the two segments of the duct separated by the acoustic source. In the segment x  L1, the acoustic pressure may be expressed as follows.

p(x, f ) A exp(ikx)  B exp(ikx)exp(i  t) L (1)

and the corresponding acoustic velocity is given by 

167

1 p u(x, f )  i   x 1  {ikAexp(ikx)  ikB exp(ikx)} exp(i  t) i   1  {A exp(ikx)  B exp(ikx)} exp(i  t)  c  c u(x, f )  {A exp(ikx)  B exp(ikx)} exp(i  t) L (2)



168

where  - is the angular frequency;   2  f  i    - is the damping coefficient (sec-1) k - is the acoustic wave number; k = /c c - is the speed of sound f - is the acoustic frequency p - is the acoustic pressure u - is the acoustic particle velocity x - is the location from the termination #1 of the duct. A - is the amplitude of the forward propagating wave B - is the amplitude of the backward propagating wave

The acoustic impedance at x=0 is defined as   p(0, f ) A  B R1( f ) 1 Z1( f )        (3)  c u(0, f ) A  B R1( f ) 1

From equation (3), the reflection coefficient, R1( f ) , can be expressed in terms of the impedance, Z1(f), as follows:   A Z1( f ) 1  R1( f )      L (4) B Z1( f ) 1  The transferred acoustic impedance, at x=L1, from termination #1 is given by

  1 p(L1, f ) A exp(ikL1)  B exp(ikL1) R1( f ) exp(ikL1)  exp(ikL1)  ZS ( f )      L (5)  c u(L1, f ) {A exp(ikx)  B exp(ikx)} R1( f ) exp(ikL1)  exp(ikL1)

In the region, L1  x  L, the acoustic pressure and acoustic velocity are given by  p(x, f )  C exp{ik(x  L)}  D exp{ik(x  L}exp(i  t) L (6)  c u(x, f )  C exp{ik(x  L)}  D exp{ik(x  L)}exp(i  t)

or  c u(x, f )  C exp{ik(L  x)}  D exp{ik(L  x)}exp(i  t) L (7)

Where L = L1 + L2 , the length of the duct. C and D are the amplitudes of the forward and backward propagating waves in segment #2 of the duct.  The acoustic impedance, Z2(f) at x = L, is defined as follows

p(L, f ) C  D 1 R2 ( f ) Z2( f )    (8)  c u(L, f ) C  D 1 R2 ( f ) or

D Z2 ( f ) 1 R2( f )   (9) C Z2 ( f ) 1

169 

The transferred impedance in the source plane from the termination #2 is given by:

  2 C exp(ikL2)  D exp(ikL2) exp(ikL2)  R2( f )exp(ikL2) ZS ( f )     (10) C exp(ikL2)  D exp(ikL2)  exp(ikL2)  R2( f )exp(ikL2)

The boundary conditions to be satisfied across the velocity source are:

 (i) Continuity of acoustic pressure

(ii) Conservation of mass

Continuity of acoustic pressure in the source plane yields

B {R1( f ) exp(ikL1)  exp(ikL1)}  C {R2 ( f ) exp(ikL2 )  exp(ikL2)} (11) from this we have C  ( f ) B (12) where R ( f ) exp(ikL )  exp(ikL )  ( f )   1 1 1  (13) R2( f ) exp(ikL2)  exp(ikL2 )

From the conservation of mass condition, we get

 C {exp(ikL2)  R2( f ) exp(ikL2)}  c uS ( f )  B{R1( f ) exp(ikL1)  exp(ikL1)} (14)

Combining (5), (10), (11) and (14) gives

 {exp(ikL2)  R2 ( f ) exp(ikL2 )}

{exp(ikL2)  R2 ( f ) exp(ikL2 )} c u ( f ) {R ( f ) exp(ikL )  exp(ikL )}  S  1 1 1 B{R1( f ) exp(ikL1)  exp(ikL1)} {R1( f ) exp(ikL1)  exp(ikL1)}

1 c uS ( f ) 1 2   1 ZS ( f ) B{R1( f ) exp(ikL1)  exp(ikL1)} ZS ( f ) or    1 2  c uS ( f ) 1 1 ZS ( f )  ZS ( f )   2  1   1 2  B{R1( f ) exp(ikL1)  exp(ikL1)} ZS ( f ) ZS ( f )  ZS ( f ) ZS ( f )  or  1 2  BR 1( f ) exp(ikL1)  exp(ikL1) pS ( f ) ZS ( f ) ZS ( f )    1 2  (15) c uS ( f ) c uS ( f ) ZS ( f )  ZS ( f )

from (15) we can write

 170

 1 2   p(x, f ) ZS ( f ) ZS ( f ) p(x, f )  c  1 2   (16) uS ( f ) ZS ( f )  ZS ( f ) pS ( f ) 

1 Equation (16) applies to both regions: 0  x L1 & L1  x L. It is an expression for the forced  response of the acoustic pressure field in the duct to a velocity source placed in it. This response function 1 2 is maximized when ZS ( f )  ZS ( f ) is minimized. Thus to obtain acoustic resonance frequencies and the corresponding damping coefficients, we must solve  Z1 ( f )  Z2( f )  0 (17) S S

Thus from (5), (10) and (17), the equation to be solved for resonance/damping analysis is:

 R ( f ) exp(ikL )  exp(ikL ) exp(ikL )  R ( f )exp(ikL ) F(k)   1 1 1   2 2 2  0 (18) R1( f ) exp(ikL1)  exp(ikL1)  exp(ikL2)  R2( f )exp(ikL2)

Simplifying (18), we get

 F(k)  exp(ikL)  R1( f ) R2( f ) exp(ikL)  0 (19)



1 My colleague, Dr. Parma Mungur, derived an expression very similar to this several years ago when we both worked in the Acoustics group at GE Aircraft Engines in Cincinnati.

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Appendix III

Calibration of sense tube type acoustic transducers for high temperature applications Written by Dr. Asif Syed

#4 #1 #2 #3

Pressure Transducer

Sense Tube Wall of the duct containing fluid flow and acoustic waves

p1 p2 p3 p4

p1, p2, etc are acoustic pressures to be sensed and measured by the pressure transducers.

Figure 1. An example of the application of sense tube type transducers.

The diagram in Figure 1 shows an application in which the acoustic pressure transducers are applied to sense tubes, which continue beyond the transducer location. The total length of the sense tube is designed so that the reflected waves from the closed end of the tube are minimized due to viscous damping. The open or the sensing end of the tube is connected to the acoustic field through the walls of the duct containing fluid flow at high temperature.

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The calibrations required for a set of sense-tube type acoustic transducer systems will include the following:

1. Absolute calibration for each transducer using a piston phone

2. Relative calibrations (relative to a transducer ―#0‖ = RCN,0(f)

3. Determine the transfer function, TF(f), of each sense tube

The schematic diagram in Figure 2 shows the setup to measure the transfer function for the sense tube.

The transducer #0 is used for reference. This transducer may not be employed in a sense tube in order to preserve it from possible damage. One end of the wave tube is closed while the other is open to sense the pressure oscillation. It can be shown that the pressure response of the transducer mounted on the wave tube at location L from the closed end is given by the following frequency response function.

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L #0 Flush mounted transducer p(f) #N Closed Pressure Sensor Open

End transducer mounted on the Wave Tube End wall of the wave tube

Test setup for determining the TF(f) of a sense tube. Transducer number 0 is a reference transducer.

p ( f ) p* ( f )  N N   *   p ( f ) p  TF( f )  0 0 C(N,0)( f )

The relative Calibration is given by



V ( f ) V * ( f ) C ( f )   N N  (N,0)  *   V0 ( f ) V0 

when the transducers #N and #0 are both flush mounted at the end of the calibration apparatus. In this configuration, both transducers are subjected to identical acoustic pressure oscillations. V(f) represents the voltage signals being generated in response to the acoustic pressure oscillation.

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