Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Heydarlaki, Ramin
Master of Applied Science - MASc
University of British Columbia
Thermoacoustic (pressure and heat release rate) oscillations can be detrimental to the operation of engineering equipment ranging from large-scale gas turbine engines (hundreds of mega watts) to small-scale power generators (few kilo watts). The focus of this thesis is thermoacoustic oscillations occurring in the combustor of a small-scale power generator. In this combustor, circulation of exhaust gas around reacting gas is utilized to increase the combustor thermal efficiency and is facilitated by using a stagnation wall. Two experimental setups are utilized. The former is a close replication of the engineering equipment, and the latter is a simplified version of the equipment, but it provides optical accessibility. In the experiments, the fuel-air equivalence ratio and air flowrate are varied from 0.7to 1.4 and 50 to 180 SLPM, respectively. For the experiments related to the first setup, pressure and temperature measurements are performed. For the second setup, both pressure and flame chemiluminescence measurements are acquired. For the first setup, the characteristics of thermoacoustic oscillations while the wall-temperature increases and how the initial wall-temperature influences these characteristics are investigated. It is shown, that for cold-start conditions, the pressure transitions from chaotic to limit cycle, chaotic, bursting, and then limit cycle oscillations. For warm-start conditions, however, the pressure features a combination of both large- and small-amplitude limit cycle oscillations. For the second setup, contributions of different types of thermoacoustic modes to Rayleigh gain is investigated. Results show that, for fuel-air equivalence ratio of 0.7, thermoacoustic oscillations feature natural structural (21 Hz) and acoustic (61 Hz) modes. However, oscillations related to fuel-air equivalence ratio of 0.9 feature intrinsic modes near 240 and 290 Hz, in addition to structural and acoustic modes. It is demonstrated that acoustic and structural modes mostly contribute to the growth of the thermoacoustic oscillations. However, generally, intrinsic modes contribute to mitigation of the thermoacoustic oscillations by providing negative acoustic energy source term mainly near the flame holder and stagnation wall. To the best knowledge of the author, the contributions of structural and intrinsic modes to Rayleigh gain have not been investigated in the past.
Experimental study of thermoacousticoscillations inside a small-scale powergenerator combustorbyRamin HeydarlakiA thesis submitted in partial fulfillment ofthe requirements for the degree ofMASTER OF APPLIED SCIENCEinThe College of Graduate Studies(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)September 2020 Ramin Heydarlaki, 2020The following individuals certify that they have read, and recommend to the College ofGraduate Studies for acceptance, a thesis/dissertation entitled:Experimental study of thermoacoustic oscillations inside a small-scale power generator com-bustorsubmitted by Ramin Heydarlaki in partial fulfillment of the requirements of the degree ofMaster of Applied ScienceDr. Sina Kheirkhah, UBC, School of EngineeringSupervisorDr. Joshua Brinkerhoff, UBC, School of EngineeringSupervisory Committee MemberDr. Patrick Kirchen, UBC, Mechanical EngineeringSupervisory Committee MemberDr. Jonathan Holzman, UBC, School of EngineeringUniversity ExamineriiAbstractThermoacoustic (pressure and heat release rate) oscillations can be detrimental to the oper-ation of engineering equipment ranging from large-scale gas turbine engines (hundreds of megawatts) to small-scale power generators (few kilo watts). The focus of this thesis is thermoacous-tic oscillations occurring in the combustor of a small-scale power generator. In this combustor,circulation of exhaust gas around reacting gas is utilized to increase the combustor thermalefficiency and is facilitated by using a stagnation wall.Two experimental setups are utilized. The former is a close replication of the engineeringequipment, and the latter is a simplified version of the equipment, but it provides optical acces-sibility. In the experiments, the fuel-air equivalence ratio and air flowrate are varied from 0.7to 1.4 and 50 to 180 SLPM, respectively. For the experiments related to the first setup, pres-sure and temperature measurements are performed. For the second setup, both pressure andflame chemiluminescence measurements are acquired. For the first setup, the characteristicsof thermoacoustic oscillations while the wall-temperature increases and how the initial wall-temperature influences these characteristics are investigated. It is shown, that for cold-startconditions, the pressure transitions from chaotic to limit cycle, chaotic, bursting, and then limitcycle oscillations. For warm-start conditions, however, the pressure features a combination ofboth large- and small-amplitude limit cycle oscillations. For the second setup, contributions ofdifferent types of thermoacoustic modes to Rayleigh gain is investigated. Results show that, foriiiAbstractfuel-air equivalence ratio of 0.7, thermoacoustic oscillations feature natural structural (21 Hz)and acoustic (61 Hz) modes. However, oscillations related to fuel-air equivalence ratio of 0.9feature intrinsic modes near 240 and 290 Hz, in addition to structural and acoustic modes.It is demonstrated that acoustic and structural modes mostly contribute to the growth of thethermoacoustic oscillations. However, generally, intrinsic modes contribute to mitigation of thethermoacoustic oscillations by providing negative acoustic energy source term mainly near theflame holder and stagnation wall. To the best knowledge of the author, the contributions ofstructural and intrinsic modes to Rayleigh gain have not been investigated in the past.ivLay SummaryThe pressure and heat release oscillations inside the combustor of a ∼ 1 kW power generatorare investigated experimentally. The generator does not always produce tolerable pressure os-cillations and sometimes generates excessive noise which could be loud. The main goals of thisstudy are to characterize the generated oscillations, identify the origins of these oscillations, andfinally explore the effects of operation conditions on growth or decay of these oscillations. Thisallows for improved design of generators which are quiet and operate durably.vPrefaceAll work presented henceforth was conducted by the author under the supervision of ProfessorSina Kheirkhah in the Combustion for Propulsion and Power Laboratory at the University ofBritish Columbia. Mr. William Aitchison and Mr. Peter Kostka from Etalim Inc. (the industrypartner) designed and manufactured the setups. Some parts of this thesis was presented in theCanadian Section of the Combustion Institute technical meeting in Kelowna in May 2019. Later,parts of the results were published in the journal of Experimental Fluid and Thermal Science[1].viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixNomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 General overview and problem statement . . . . . . . . . . . . . . . . . . . . . . 11.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Pressure and heat release rate oscillations coupling . . . . . . . . . . . . . 31.2.2 Thermoacoustic modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7viiTABLE OF CONTENTS1.2.3 Thermoacoustic driving mechanisms . . . . . . . . . . . . . . . . . . . . . 131.2.4 Types of thermoacoustic oscillations . . . . . . . . . . . . . . . . . . . . . 151.3 Thesis objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Chapter 2: Experimental Methodology . . . . . . . . . . . . . . . . . . . . . . . . 202.1 Experimental methodology related to the setup with the metallic combustor . . . 202.1.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.2 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.3 The tested conditions and the stability map . . . . . . . . . . . . . . . . . 232.2 Experimental methodology related to the optically accessible setup . . . . . . . . 262.2.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.2 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.3 Data reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.4 Tested conditions and the flame configurations . . . . . . . . . . . . . . . 31Chapter 3: The effect of wall-temperature on thermoacoustic oscillations . . . 353.1 Pressure oscillations behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Long time-period characteristics of the pressure oscillations . . . . . . . . . . . . 383.3 Transient behavior of the pressure oscillations for cold-start conditions . . . . . . 413.4 Transient behavior of the pressure oscillations for warm-start conditions . . . . . 473.5 Relations between spatially averaged Helmholtz and Strouhal numbers versus theReynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Chapter 4: Contributions of structural, acoustic, and intrinsic modes to Rayleighgain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55viiiTABLE OF CONTENTS4.1 Characterization of structural and acoustic oscillations . . . . . . . . . . . . . . . 554.2 Pressure and spatially averaged flame chemiluminescence oscillations characteristics 574.2.1 Non-reacting flow dominant modes . . . . . . . . . . . . . . . . . . . . . . 574.2.2 Reacting flow dominant modes . . . . . . . . . . . . . . . . . . . . . . . . 584.3 Spatial relevance of flame chemiluminescence to thermoacoustic modes . . . . . . 614.4 Relations between the Rayleigh gain and pressure oscillations amplitude . . . . . 644.5 Critical regions related to the structural, acoustic, and intrinsic thermoacousticmodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Chapter 5: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.1 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Appendix A:Thermoacoustic oscillations characteristics during flame shapetransition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91A.1 Temporal and spectral thermoacoustic characteristics during flame shape transition 91A.2 Characteristics of main and exhaust tubes flame chemiluminescence . . . . . . . 94ixList of TablesTable 2.1 Tested experimental conditions. The asterisk sign refers to warm-startcondition. Tw,i is the initial combustor wall-temperature. . . . . . . . . . . 24Table 2.2 Tested conditions with the fuel-air equivalence ratio set to 0.7 . The unitof Q˙ is in Standard Liters Per Minute. . . . . . . . . . . . . . . . . . . . . 32Table 2.3 Tested conditions with the fuel-air equivalence ratio set to 0.9 . The unitof Q˙ is in Standard Liters Per Minute. . . . . . . . . . . . . . . . . . . . . 32Table 3.1 The distances in Fig. 3.6(a). The values are in millimetres. . . . . . . . . 51xList of FiguresFigure 1.1 Schematic of a reverse-flow configuration utilized to preheat the reactants. 2Figure 1.2 Schematic of a tube with a constant cross sectional area. . . . . . . . . . 8Figure 1.3 Schematic of a Helmholtz resonator. Gas in the main body and that inthe neck of the bottle serve as spring and mass, respectively. . . . . . . . 10Figure 1.4 Schematic of a Helmoltz resonator in combustors with inlet and outlet flow. 11Figure 1.5 Schematic of a feedback mechanism in intrinsic modes. . . . . . . . . . . 11Figure 1.6 Schematic of the coupling between gas pressure oscillation and combustorwall vibrations in natural structural modes. . . . . . . . . . . . . . . . . . 13Figure 2.1 (a) The first experimental setup, (b) top view of the preheating/premixingtube, and (c) schematic of the combustion chamber and the preheat-ing/premixing tube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22xiLIST OF FIGURESFigure 2.2 Stability map of the power generator shown in Fig. 2.1. Cells with bluecolour pertain to conditions that the mixture was ignited and the flamewas stable. For the conditions with the yellow colour, the mixture wasignited, but it was extinguished after a few seconds. Conditions withpink colour pertain to those that flame could only be stabilized if theburner was warmed-up initially. The mixture cannot be ignited even ifthe burner was warmed-up for conditions denoted by the red colour. Notethat conditions with the asterisk sign are associated with the warm-start,i.e. Tw,i = 680 K. The conditions without the asterisk refer to Tw,i = 290 K. 25Figure 2.3 Variations of the fuel (a) and air (b) flowrates for condition B −D. Thefuel-air equivalence ratio is kept constant and equal to 0.9 throughout theexperiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 2.4 (a and b), (c), and (d) are the technical drawings and schematic of theexperimental setup, detailed drawing of the burner, and the utilized tur-bulence generator, respectively. . . . . . . . . . . . . . . . . . . . . . . . . 27Figure 2.5 (a) Raw, (b) background subtracted and sensitivity corrected (c) binned,and (d) median-based filtered images. . . . . . . . . . . . . . . . . . . . . 30Figure 2.6 The alphabets in (a), and (b) pertain to the tested conditions presentedin Tables 2.2 and 2.3, for φ = 0.7 and 0.9, respectively. White andpink background colours correspond to Bunsen and M-shaped flames,respectively. D2 experimental condition can feature both Bunsen andM-shaped flames. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33xiiLIST OF FIGURESFigure 2.7 (a) and (b) are the time-averaged OH∗ chemiluminescence images relatedto Flames D1 and D2, pertaining to Bunsen and M-shaped flames re-spectively. The dashed-ellipses highlight the exterior wings of the flamesformed on the torus-shaped flame-holder. The white dashed-region in (b)highlights the flame formed on the igniter rod. . . . . . . . . . . . . . . . 34Figure 3.1 (a) Variation of the pressure oscillations for condition A∗. (b) and (c)are insets of (a). (d), (e), and (f) are samples of the pressure oscillationspertaining to conditions B, C∗, and B–D, respectively. The blue curvesare the instantaneous pressure oscillations amplitude. . . . . . . . . . . . 36Figure 3.2 Representative normalized power spectrum density of the pressure oscil-lations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 3.3 Representative normalized probability density functions of (a) the pres-sure oscillations and (b) the pressure oscillations amplitude. . . . . . . . 40Figure 3.4 (a) Combustor wall-temperature variation. The black solid and dottedcurves are the experimentally measured temperature and the right-hand-side of Eq. (3.2), respectively. (b) Spectrogram of the pressure oscillations.The dotted curves in (b) are the right-hand-side of Eq. (3.4) for the threeoscillations modes. (c) Variation of the pressure oscillations PDF esti-mated for a sliding time window. (d-g), (h-k), and (l-o) are the pressuretime histories, phase-space trajectories, and the Poincare´ maps, respec-tively. The results in (d, h, l), (e, i, m), (f, j, n), and (g, k, o) pertain tot = [29 31], [302 304], [847.4 848.6], and [1482 1486] s, respectively. Theresults pertain to condition B −D (a cold-start condition). . . . . . . . . 43xiiiLIST OF FIGURESFigure 3.5 (a) Combustor wall-temperature variation with time. (b) Spectrogramof the pressure oscillations. (c) PDF of the pressure oscillations. (d, g,and j), (e, h, and k), and (f, i, and l) pertain to [0.1 3], [201 205], and[484 500] s, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 3.6 (a) The experimental schematic along with η vector, (b) Temperature spa-tial variation for η vector (c) Variation of wall-temperature and mixturetemperature versus time for fuel-air equivalence ratio of 1 . . . . . . . . . 51Figure 3.7 Variations of the Helmholtz (first row) and Strouhal (second row) numbersversus the Reynold number. The first, second, and the third columnspertain to the first, second, and the third modes, respectively. . . . . . . 53Figure 4.1 (a-f) are samples of the exhaust displacement (along three axes) due tomechanical impulse, pressure oscillations of gas inside the setup excitedby the mechanical impulse, pressure oscillations generated by speakerexcitation at 200 Hz, and vibrations produced by the speaker excitationsat 90 Hz, respectively. (g) is the power spectrum densities of the resultspresented in (a-f). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Figure 4.2 Power spectrum density of the pressure oscillations for non-reacting con-ditions. S is the location of the igniter rod. . . . . . . . . . . . . . . . . . 58Figure 4.3 Power spectrum density of the pressure oscillations (first row) and flamechemiluminescence (second row) oscillations for reacting conditions. Theresults pertain to fuel-air equivalence ratio of 0.7. Solid and dashed-curveshighlight Bunsen and M-shaped flames, respectively. . . . . . . . . . . . . 59xivLIST OF FIGURESFigure 4.4 Power spectrum density of the pressure (first row) and flame chemilu-minescence (second row) oscillations at fuel-air equivalence ratio of 0.9.Solid curves pertain to Bunsen and dashed-curves pertain to M-shapedflames. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Figure 4.5 Flame chemiluminescence P˜SD for Q˙air = 120 SLPM. The first and sec-ond rows pertain to P˜SD evaluated near f = fs and f = fac for φ = 0.7,respectively. The third, fourth, and fifth rows pertain to P˜SD evaluatedat f = fs, fa, and fITA,1, respectively. A1–A4 and E1–E4 are the testedconditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Figure 4.6 Flame chemiluminescence P˜SD for Q˙air = 180 SLPM. The first and sec-ond rows pertain to P˜SD evaluated near f = fs and f = fac for φ = 0.7,respectively. The third, fourth, fifth, and sixth rows pertain to P˜SD eval-uated at f = fs, fa, fITA,1, and fITA,2, respectively. D1–D4 and H1–H4are the tested flames conditions. . . . . . . . . . . . . . . . . . . . . . . . 63Figure 4.7 (a) and (b) are the spatially and temporally averaged Rayleigh gain forthe thermoacoustic oscillations band-pass filtered around f = fs and fa,respectively. (c) is the summation of results in (a) and (b). (d) is thevariation of the pressure fluctuations root mean square, and (e) is varia-tion of the spatially and temporally averaged Rayleigh gain versus p′rms.The results pertain to φ = 0.7. . . . . . . . . . . . . . . . . . . . . . . . . 65xvLIST OF FIGURESFigure 4.8 (a), (b), (c), and (d) are the spatially and temporally averaged Rayleighgain for the thermoacoustic oscillations band-pass filtered around f = fs,f = fac, f = fITA,1, and fITA,2, respectively. (e) presents summation ofresults in (a–d). (f) is the variation of the pressure fluctuations root meansquare, and (g) is the variation of the spatially and temporally averagedRayleigh gain versus p′rms. The results pertain to φ = 0.9. . . . . . . . . . 66Figure 4.9 The first and second rows are ψ related to fs and fa for φ = 0.7, re-spectively. The third, fourth, and fifth rows pertain to ψ evaluatednear fs, fa, and fITA,1, respectively, for φ = 0.9. The results pertainto Q˙air = 120 SLPM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Figure 4.10 The first and second rows are ψ evaluated near fs and fa for φ = 0.7,respectively. The third, fourth, fifth, and sixth rows pertain to ψ evalu-ated near fs, fa, fITA,1, and fITA,2, respectively, for φ = 0.9. The resultspertain to Q˙air = 180 SLPM. . . . . . . . . . . . . . . . . . . . . . . . . . 68Figure A.1 (a) and (d) pertain to spatially averaged flame chemiluminescence in themain tube and pressure oscillations related to FlameD2 condition, respec-tively. (b) and (c) are OH∗ chemiluminescence at t = 3.09 and 4.09 s,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Figure A.2 (a) and (b) are the pressure oscillations spectrogram and power spectrumdensity for Flame D2 condition, respectively. . . . . . . . . . . . . . . . . 93Figure A.3 (a) and (b) are the spatially averaged OH∗ chemiluminescence signal in-side the main and one exhaust tube, respectively. The results pertain toFlame D1 condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95xviLIST OF FIGURESFigure A.4 (a) and (b) are the spatially and temporally averaged flame chemilumi-nescence in the main and exhaust tubes, respectively. . . . . . . . . . . . 95xviiAcknowledgementsFirst, I would like to express my deepest gratitude to my supervisor, Dr. Sina Kheirkhah, forgiving me the opportunity to learn and grow. His resourceful sense of criticism while dealing withproblems and potential solutions changed my mindset from “how to find an answer” to “how tofind the best answer”. This project would not have been possible without his encouragement,patience, and his dedication in training me to become a disciplined researcher.I am truly thankful to Messrs. William Aitchison and Peter Kostka for manufacturing of thesetups and the measurement campaigns hold at UBC Okanagan. My sincerest gratitude to UBCOkanagan School of Engineering staff, especially, Tim Giesbrecht, Raymond Seida, and DavidZinz for always helping me out and lending me tools and equipment. Next, I want to thank allof my co-workers and friends at UBCO. All of you made the last two and half years of my life anot-repeatable and enjoyable experience. Looking back to some of the activities we did togetherbrings me joy. From Sajjad’s soccer games to Arash’s late night hangouts. From Morales’ssafety lectures to his 4.5 minutes tea breaks. From Sahar’s help performing the experiments toDeepanshi’s look to my setup like it was never going to work. I will never forget these momentsand I thank you all for creating them with me. Last but not least, I thank my family and friendsfor always being with me from long distances all around the world.Finally, I would like to acknowledge the Natural Sciences Engineering Research Council(NSERC) of Canada and Etalim Inc. for their financial support during my study.xviiiDedicationTo my mom and grandmaxixNomenclatureSymbolsAi Inlet cross section area in Helmholtz resonatorsAn Neck cross section area in Helmholtz resonatorsc Speed of soundc Spatially averaged speed of soundD Diameter of the bluff-body used to stabilize the flamesD Acoustic energy dissipation per unit volumed DisplacementE Acoustic energy per unit volumef Frequencyfa Acoustic mode frequencyfi Frequency of the ith modefITA,1 First intrinsic thermoacoustic mode frequencyfITA,2 Second intrinsic thermoacoustic mode frequencyfs Natural structural mode frequencyf∞,i Frequency of the ith mode at T = T∞g Gravitational accelerationH Hilbert transformxxNOMENCLATUREHe Helmholtz numberL Length of the combustion chamberl Total length of the setupl1−10 Geometrical distancesli Length of the inlet neck in combustorsln Length of the neck in Helmholtz resonatorslo Length of the outlet neck in combustorslr Length of the preheating/premixing sectionn Integer numberOH∗ Flame chemiluminescence used to study heat release rateOH∗M (t) Spatially averaged flame chemiluminescence in the main tubeOH∗E (t) Spatially averaged flame chemiluminescence in the exhaust tube〈OH∗〉MSpatially and temporally averaged flame chemiluminescence inside the maintube〈OH∗〉ESpatially and temporally averaged flame chemiluminescence inside the ex-haust tubep Pressurep¯ Temporally averaged pressurep′ Presssure oscillationsp˜′ Presssure oscillations amplitudep′rms Pressure oscillations root mean squareQ˙air Air volumetric flowrateQ˙CH4 Methane volumetric flowratexxiNOMENCLATUREq˙′ Heat release rate oscillationsRe Reynolds numberS Igniter rod tip positionSt Strouhal numberT One period of the oscillationsTad Adiabatic flame temperatureTc Cold reactants temperatureTe Exhaust gas temperatureTr Warm reactants temperatureTw Combustor wall-temperatureTw,i Inital combustor wall-temperaturet Timeti Time related to the start of ith cycleU Mean bulk flow velocity−→u′ Velocity fluctuations vectorV Volumex Position vectorx Streamwise directiony Axis normal to the streamwise directionGreek symbolsγ Gas heat capacity ratioη Integration variableν Gas kinematic viscosityxxiiNOMENCLATUREν Spatially averaged kinematic viscosityτ Time lag used to obtain Poincare´ mapsτin Integration variableφ Fuel-air equivalence ratioψ Rayleigh gainψ Temporally averaged Rayleigh gainψ Spatially and temporally averaged Rayleigh gainAbbreviationsDAQ Data acquisitionFOV Field of viewITA Intrinsic thermoacousticPDF Probablity density functionPDF ∗ Normalized probability density functionPSD Power spectrum dendityPSD∗ Normalized power spectrum densityP˜SD Conditioned power spectrum densitySLPM Standard liters per minutexxiiiChapter 1Introduction1.1 General overview and problem statementThis thesis is concerned with thermoacoustic (pressure, p′, and heat release rate, q˙′) oscil-lations of a portable power generation unit combustor. The unit is composed of two sections,an acoustic generation section (the combustor) and an acoustic to electric energy conversionsection. Depending on the operation conditions, the combustor may generate relatively large-amplitude thermoacoustic oscillations. While the small amplitude thermoacoustic oscillationsare desired for the operation of the unit and conversion of acoustic energy to electricity, largeamplitude thermoacoustic oscillations are detrimental to the operation of the unit and couldpotentially lead to mechanical fatigue [2–5]. How thermoacoustic oscillations are converted toelectric energy is not the focus of this thesis. Instead, experiments and analyses are performed tounderstand and characterize the large amplitude thermoacoustic oscillations. Later, operationguidelines and future design specifications are recommended so that thermoacoustic oscillationsamplitude is reduced to a certain level such that, first, the whole unit operates smoothly, and,second, enough pressure oscillations are provided for sustainable operation of the acoustic toelectric energy conversion section.The combustor utilizes a reverse-flow configuration, recirculating the exhaust gas using astagnation wall, mimicking an external heat exchanger as shown schematically in Fig. 1.1. In11.1. General overview and problem statementFig. 1.1, after reactants (green arrows) enter the plenum, they are preheated by the exhaust gas(red arrows). The heat transfer from exhaust gas to reactants is highlighted by orange arrowsin Fig. 1.1.Fig. 1.1. Schematic of a reverse-flow configuration utilized to preheat the reactants.Although the exhaust gas recirculation is desired for improving the heat exchanger ther-mal efficiency, it causes two issues. First, the stagnation wall-temperature changes during theoperation, which changes the amplitude of the thermoacoustic oscillations. The effect of thecombustor wall-temperature on the thermoacoustic oscillations are not well-understood both inthe literature and in our setup. Second, once the combustor is lit, it features mechanical vibra-tions. How and if these vibrations are coupled with the thermoacoustic oscillations are also notwell-understood both in the literature and in our setup.In terms of operation, emissions legislation requires portable power generators to operateunder lean conditions (fuel-air equivalence ratio <1) to decrease the combustion products tem-perature, and as a result, reduce NOx generation and emission [6, 7]. The industrial apparatusused in this thesis operates under lean conditions.21.2. Background1.2 BackgroundLiterature related to coupling between pressure and heat release rate oscillations, couplingmodes, mechanisms driving the the oscillations, and the types of the oscillations are brieflyreviewed here.1.2.1 Pressure and heat release rate oscillations couplingA framework that allows for quantifying the coupling between the heat release rate andpressure oscillations was first introduced by Rayleigh [8]. Later, Candel [9] explained the roleof pressure and heat release rate oscillations coupling in the context of acoustic energy (energycarried out by the traveling waves) generation through a simplified transport equation for specific(per unit volume) acoustic energy (E ), which is given by:∂E (x, t)∂t+ ∇ ·[p′(x, t)−→u′ (x, t)]=γ − 1γp¯(x)[p′(x, t)q˙′(x, t)]+D(x, t), (1.1)where x, t,−→u′ , γ, p¯(x), and D(x, t) are the position vector, time, velocity fluctuations vector,the gas heat capacity ratio, temporally averaged pressure, and the specific acoustic energydissipation, respectively. The two terms on the left hand side of Eq. (1.1) are the rate ofchange of the specific acoustic energy with respect to time and the specific acoustic energy flux,respectively. On the right hand side of the Eq. (1.1), the first term is the source of the specificacoustic energy due to the coupled heat release rate and pressure fluctuations. The acousticenergy increases if the source term exceeds the specific acoustic energy dissipation. In manyengineering applications, including that of the present study, the flames are acoustically compact.This means that, over the region that the heat release rate changes spatially, the pressure doesnot. As a result, for such applications, p′(x, t) ≈ p′(t). For acoustically compact flames, it can31.2. Backgroundbe shown that the source term is proportional to the multiplication of the pressure, p′(t), andlocal heat release rate, q˙′(x, t), oscillations. The source term integrated over ith period of theoscillations (T ) is defined as the Rayleigh gain (ψi) [10–12], and is given byψi(x, ti) =∫ ti+Ttip′(τin)q˙′(x, τin)dτin, (1.2)where τin is an integration variable and ti corresponds to the start of ith cycle. Positive (negative)ψ(x, ti) is considered to contribute to growth (decay) of the thermoacoustic oscillations, whendamping is negligible [10, 13–17].In order to identify the locations inside the combustion chamber whose pressure and heatrelease rate oscillations coupling leads to growth/mitigation of the thermoacoustic oscillations,previous investigations [13–17] estimated the Rayleigh gain. In these studies [13–17], generally,the positive (negative) Rayleigh gain pertains to a local phase difference between the pressureand heat release rate oscillations ranging between 0 and π/2 (π/2 and π). In the studies of [13–21], the Rayleigh gain (heat release rate and pressure oscillations coupling averaged over onepressure cycle) and the dynamics of the pressure and heat release rate coupling (instantaneousmultiplication of heat release rate and pressure oscillations) were investigated for swirl-stabilized[13–19] and bluff-body stabilized [20, 21] flames, with the related findings discussed below.For swirl-stabilized flames, the Rayleigh gain is dependent on the swirl number, turbulenceintensity, and flame location [15–17]. The swirl number, which is defined [22] as the ratio ofmean tangential to axial velocity, plays a prominent role in generation of positive and negativeψ regions [13–16, 18, 19]. Specifically, for small swirl numbers (less than 0.7), study of Leeet al. [16] shows that shear layers feature negative ψ, but the inner and outer recirculationzones (generated by the swirler) feature positive Rayleigh gain [15, 16]. Mahesh et al. [15]41.2. Backgroundsuggest that the positive Rayleigh gain in the outer recirculation zone is linked to the interactionbetween flames generated around the dump/injection plane and flames on top of (far from) thedump plane that are bent toward the outer recirculation zones. This interaction causes periodicgeneration and destruction of flame surfaces which are in-phase with the pressure oscillations,leading to positive ψ [15]. They [15] suggest that rotating the swirler, which is accompanied byincrease of the swirl number and the flow turbulence intensity, weakens the outer recirculationvortices, decreasing the positive value of ψ. Study of Temme et al. [17] shows that the Rayleighgain strongly depends on the flame location. They [17] showed that the tested flames maybe either lifted from or attached to the combustor dump plane. They [17] conditioned theRayleigh gain based on the flame location, and their conditionally averaged results show thatthe positive Rayleigh gain corresponds to the instants that the flame is positioned near theinjection plane. However, the negative Rayleigh gain pertains to the instants that the flame islifted and positioned farther downstream [17].Dynamics of the regions that feature a positive/negative source term (instantaneous pressureand heat release rate multiplication) has been investigated for both swirled-stabilized [17–19]and bluff-body stabilized [20, 21, 23] flames. The dynamics can be dependent on the modeof combustion, the pressure oscillations amplitude, fuel-air equivalence ratio, and the ignitionlocation [18–21, 23]. During small amplitude pressure oscillations, bluff-body stabilized flamesin the study of Krishnan et al. [21] generate small island-like regions with a positive sourceterm which are surrounded by regions with a negative source term. However, during largeamplitude pressure oscillations, large regions with a positive source term can be observed aroundcoherent flow structures that are formed downstream of the bluff-body. For such large amplitudeoscillations and as the pressure oscillations amplitude increases, the coherent structures that are51.2. Backgroundinitially formed at the combustor dump plane convect downstream and grow in size [21]. It isshown in Krishnan et al. [21] that the source term is maximized when the pressure oscillationsare maximized and the coherent structures (which contain strong positive source term) impingeon the side-wall of the combustor [21].For an aeronautical liquid-fueled combustor, Kheirkhah et al. [18] showed that, when thepressure oscillations amplitude is nearly constant, the regions that feature positive source termmay move towards the burner exit through shear layers and/or toroidal-shape structures. They[18] show that during the periods that the oscillations amplitude increases, the regions withpositive source term move upstream towards the burner exit plane. Opposite behavior is reportedin [18] when the pressure oscillations amplitude decreases [18]. Mondal et al. [20] show thatat stoichiometric condition, the combustion chamber contains regions with both positive andnegative source term. They [20] show that, decreasing the fuel-air equivalence ratio from unity to0.6, the regions with negative source term disappear leading to large amplitude thermoacousticoscillations.Results of Schuller et al. [23] show that the ignition location can influence pressure and heatrelease rate coupling through flame shape change. They [23] investigated the thermoacousticoscillations of both a laminar Bunsen as well as a V-shaped flame interacting with a Bunsenflame (referred therein as M-shaped flame). The V-shaped flames were stabilized using a centralbluff-body. Both laminar Bunsen and M-shaped flames featured constant amplitude pressureand heat release rate oscillations; however, differences were observed depending on the ignitionlocation. Schuller et al. [23] show that if the ignition source is positioned close to the burner rim,Bunsen flames are formed; otherwise, if it is positioned near the central bluff-body, M-shapedflames are formed. On one hand, Candel and co-workers [23, 24] show that, for M-shaped flames,61.2. Backgroundthe interaction of the constituent V-shaped and Bunsen flames leads to flame surface destruction,which causes reduction of the heat release rate. However, on the other hand, during the Bunsenand V-shaped flame interactions, pockets of reactants are pinched-off of the flames and convectdownstream [23, 25]. Later, these pockets burn, increasing the heat release rate. The heat releaserate reductions and generations are in-phase with the pressure oscillations leading to positivecoupling between pressure and heat release rate of the laminar M-shaped flames [23, 24]. Schulleret al. [23] argued that similar scenario happens for Bunsen flames; however, the flame surfacedestruction and reactants pocket formation is less pronounced for the Bunsen flames. As aresult, their tested Bunsen flames featured relatively weak thermoacoustic oscillations comparedto those of the M-shaped flames [23].1.2.2 Thermoacoustic modesThermoacoustic modes are identified using the power spectrum densities (PSDs) of thepressure and heat release rate oscillations [12, 26, 27]. These modes [28] can be categorizedinto acoustic [17, 26, 27, 29–32], natural structural [2, 26], and intrinsic [33–40] modes, with thecorresponding background discussed in the following.Acoustic modesGenerally, acoustic modes can be categorized into those pertaining to standing pressure wavesor those associated with the movement of air (and as a result oscillations of pressure) inside thecombustion chamber [41, 42]. The former category is referred to as the fundamental modes,[26, 30, 40, 43, 44] and the latter category is referred to as the Helmholtz modes [17, 23, 45–47].Fundamental modes are dependent on the acoustic reflections (boundary conditions) at the inletand outlet of the system. Figure 1.2 presents schematic of a tube, with acoustic boundaries at71.2. Backgroundthe inlet/outlet being open/open or open/closed. The former and the latter boundary conditionslead to generation of half and quarter acoustic waves, respectively.For open/open and closed/open boundary conditions, the fundamental frequencies and theirharmonics (fa) can be estimated from Eqs. (1.3) and (1.4), respectively, which are given by:Fig. 1.2. Schematic of a tube with a constant cross sectional area.fa = nc2L, n ∈ N, (1.3)fa = (2n− 1) c4L, n ∈ N, (1.4)where n, L, and c are integer numbers, length of the tube, and the speed of sound insidethe tube, respectively. Equations (1.3) and (1.4) can allow for predicting the fundamentalacoustic frequencies of combustion chambers with a simple geometry (such as that shown inFig. 1.2). However, for combustion chambers with complex geometries, finite element methods[26, 42], acoustic network modeling [26, 28, 37, 40], non-reacting flow experiments [26], and forcedacoustic excitations [23, 26] are used in the literature to estimate fa. The estimated acousticmode frequencies can be associated with either the entire setup [26, 37, 40], combustion chamber[2, 27, 28, 31], and/or combination of different components of the setup [42]. For example,Fritsche et al. [42] reported several modes in their pressure oscillations related to the acousticmodes of their entire system, combination of plenum and a swirl-inducing burner (positioneddownstream of the plenum and upstream of the combustion chamber), as well as combination81.2. Backgroundof the combustion chamber and the burner. Although fa estimated using the above methods donot always match the prediction of Eqs. (1.3) and (1.4), trends related to the effects of speed ofsound and length of the combustion chamber (or lengths of the other parts of the setup) followthose suggested by these equations. For example, in the studies of [26, 31, 48, 49], the acousticmode frequency decreases by increasing the combustion chamber length, which agrees with theprediction of Eqs. (1.3) and (1.4). Increasing the combustor thermal power (by increasing thefuel-air equivalence ratio of lean flames or increasing the mean bulk flow velocity) increases thecombustion products temperature, the speed of sound, and as a result, the fundamental acousticfrequency [26, 43, 50]. Although the above trends agree with the predictions of Eqs. (1.3) and(1.4), discrepancies are observed. For example, in some investigations [26, 40, 50], it is reportedthat changing the combustor thermal power does not change fa and the oscillations lock ontoa fixed frequency. For example, despite the reported [50] linear variation of fa with the meanbulk flow velocity for lean flames, for rich flames, the study of Samaniego et al. [50] show that faremains unchanged by increasing the mean bulk flow velocity. This is related to the differencein the regions that drive (positive ψ) the oscillation related to the rich and lean flames acousticmodes. For lean (rich) flames, these regions are close to (far from) the combustor dump plane[50]. It is speculated by the author that increasing the mean bulk flow velocity mostly increasesthe temperature in regions close to the dump plane, and as a result, the acoustic frequency ofthe lean flame increases. Nevertheless, the amplitude of the acoustic modes generally increasesby increasing mean bulk flow velocity [26, 30, 40, 43, 44].The second type of the acoustic modes is referred to as the Helmholtz mode [45–47]. Thismode appears when part of the setup serves as a resonator. As schematically shown in Fig. 1.3,the gas inside the neck of the bottle-shaped setup oscillates because of the flow perturbations91.2. Backgroundoutside of the bottle. This is usually mimicked by a mass and a mass-less spring system, inwhich gas in the body of the bottle and that in the neck of the bottle serve as the spring andmass, respectively [45–47]. The frequency of the mass oscillations can be estimated usingFig. 1.3. Schematic of a Helmholtz resonator. Gas in the main body and that in the neck of thebottle serve as spring and mass, respectively.fac =c2π√AnV ln, (1.5)where An, V, and ln are the neck cross section area, volume of the gas trapped in the mainbody, and length of the neck, respectively. Helmholtz resonators can model pressure oscillationsinside combustors with inlet and outlet flow as shown in Fig. 1.4. In the figure, Ai, li, and lo arethe inlet area cross section, length of the inlet neck, and length of the outlet neck, respectively.For complex combustor geometries, a combination of these parameters are used to calculate Anand ln in Eq. (1.5) and as a result fa [17, 31, 51]. In addition to Eq. (1.5), numerical simulations,e.g. acoustic network modeling, can be used to predict the Helmholtz modes frequencies [28,31, 37]. Usually, identifying the Helmholtz mode frequency is not straightforward for complexcombustion chamber geometries. For example, in the study of Temme et al. [17], V , in Eq. (1.5),is taken only as the combustion chamber volume and does not include that of the plenum. Infact, they [17] observed changing the plenum volume does not change the Helmholtz frequency.The authors [17] observed two Helmholtz modes around 80 and 160 Hz which are respectivelyrelated to the small and large area inlet orifices in their setup.101.2. BackgroundFig. 1.4. Schematic of a Helmoltz resonator in combustors with inlet and outlet flow.Intrinsic thermoacoustic modesGenerally, acoustic modes arise as a result of acoustic reflections from the system boundaries[4, 39] and/or Helmholtz resonator-type pressure oscillations [28, 31, 37]. However, the intrinsicthermoacoustic (ITA) modes oscillations exist due to a feedback mechanism between pressureand heat release rate oscillations within the flame [33, 34, 52]. Such flame-intrinsic feedbackmechanism creates an instability which is different from hydrodynamic and thermodiffusiveinstabilities [33]. As schematically shown in Fig. 1.5, for intrinsic thermoacoustic oscillations,the flame generates acoustic perturbations (solid red arrow) which cause velocity fluctuationsupstream of the flame in the inlet flow. Then, the velocity fluctuations lead to heat release rateoscillations (dashed red arrow) [36, 37].Fig. 1.5. Schematic of a feedback mechanism in intrinsic modes.Intrinsic modes are studied in semi-anechoic (absence of acoustic reflections) setups numer-ically [33, 37, 39] and experimentally [38, 40]. Anechoic conditions are used as they facilitatemitigation of the acoustic modes. Thus, the observed oscillations are considered to pertain to111.2. Backgroundthe ITA [33, 37–40] modes. Murugesan et al. [40] show that the frequency of intrinsic modeslinearly increases by increasing the input power of the combustor either by increasing the fuel-airequivalence ratio [38] and/or increasing the mean bulk flow velocity [40]. Since, for lean flames,increasing the fuel-air equivalence ratio decreases the flame height, the ITA modes frequenciesare inversely related to the flame height. Albayrak et al. [39] show that the ratio of ITA modesfrequencies is equal to the ratio of the tested mean bulk flow velocities raised to the power of0.8 multiplied by the inverse of the flame height ratio [39]. Studies of Albayrak et al. [39] andMurugesan et al. [40] also show that intrinsic modes are prone to exist in smaller mean bulkflow velocities (less than 11 m/s); and, for large mean bulk flow velocities, the thermoacousticoscillations tend to lock onto the acoustic modes. Compared to the combustor fundamentalacoustic frequency, that of the ITA modes is independent of the combustor length [31, 40].Natural structural modesLocking of the thermoacoustic oscillations frequencies to acoustic and intrinsic thermoa-coustic modes has been observed and reported in several past investigations [26, 30, 33, 37–40, 40, 43, 44], with the corresponding details discussed earlier. To the best knowledge of theauthor, few studies [2, 26] show, in addition to the above, the thermoacoustic oscillations mayalso lock onto the natural structural modes, with a frequency of fs. A schematic explaining cou-pling through fs is shown in Fig. 1.6. For such modes, as a first step, gas pressure oscillationscause vibrations in the combustor. In the second step as shown in Fig. 1.6, the vibrations of thecombustor wall lead to the neighboring gas displacement, and as a result, disturb the acousticfield which couples with the heat release rate oscillations [2, 26].The frequency of natural structural mode can be experimentally measured by hammering thecombustor and acquiring the wall oscillations using accelerometer and laser Doppler vibrometer121.2. BackgroundFig. 1.6. Schematic of the coupling between gas pressure oscillation and combustor wall vibra-tions in natural structural modes.for non-reacting and reacting conditions, respectively [26]. This type of coupling and how itcan contribute to growth or decay of thermoacoustic oscillations are not studied in detail in theliterature and require further investigations.1.2.3 Thermoacoustic driving mechanismsThe coupling between combustion heat release rate and pressure oscillations is the drivingmechanism for the thermoacoustic oscillations. The present study deals with gaseous-fueledcombustors. For such combustors, the oscillations are driven either by the acoustics or byshedding of coherent flow structures. For the former, as the pressure increases [8, 53], the gas inletflowrate decreases. This causes a reduction in the generated heat release rate. This is followedby reduction of pressure and increased suction of mixture into the combustion chamber, whichincreases the generated heat release rate. This phenomenon repeats, leading to self-sustainedoscillations of heat release rate and pressure. For the oscillations driven by shedding of thecoherent flow structures, three different sub-mechanisms have been reported in the literature[54]. First, coherent flow structures may interact with the premixed flame surface, modulatingthe flame surface area and strain rate, and as a result, changing the heat release rate. Second,131.2. Backgroundthe vortices can cause entrainment of the fresh air into the reactants, locally changing the fuel-air equivalence ratio, and as a result, changing the heat release rate [54, 55]. Third, impingingvortices on different parts of the combustor can cause thermoacoustic oscillations by generatinga strong and sudden heat release at the location of impingement [21, 56].Variations of the Helmholtz (He = fL/c) and the Strouhal (St = fD/U) numbers versusthe Reynolds (Re = UD/ν) number have been used to distinguish acoustic- or coherent flowstructure-driven oscillations. See for example [2, 26, 30, 57, 58]. In the definitions of He, St,and Re numbers, f , L, D, U , and ν are the frequency, length of the combustion chamber andplenum, diameter of the bluff-body used to stabilize the flames, mean bulk flow velocity (whichis related to the total fuel and air volumetric flowrate), and the reactants kinematic viscosity,respectively. Casado [26], Shahi et al. [2, 58], Alemela et al. [57], and Chakravarthy et al. [30, 59]investigated variations of St and He with the Reynolds number estimated based on the coldflow kinematic viscosity for partially-premixed and non-premixed flames of gaseous fuels. Theirresults [2, 26, 30, 57–59] show that the Helmholtz and Strouhal numbers depend on the Reynoldsnumber as well as the location of the bluff-body. Casado [26], Shahi et al. [2, 58], and Alemalaet al. [57] discuss that increasing the Reynolds number decreases the Strouhal number, but theHelmholtz number remains approximately constant. As a result, they [2, 26, 57, 58] concludedthat their combustor dynamics is dominated by the acoustics. Compared to the studies of[2, 26, 57, 58], Chakravarthy et al. [30, 59] showed that the relation betweenHe and Re numbersdepends on the distance between the stabilizing bluff-body and the inlet of the combustionchamber. For relatively small distances, the Helmholtz number is constant (about 0.6) anddoes not vary with the Reynolds number, which agrees with the observations of [2, 26, 57, 58].However, for large values of the distance between the bluff-body and the inlet to the combustor,141.2. Backgroundthe pressure oscillations frequency may lock onto the vortex-shedding frequency, featuring anearly linear relation between He and Re numbers. The lock-on phenomenon suggests thesystem dynamics is controlled by the vortex-shedding. Such behavior has been observed inseveral instances related to thermal fluids such as flow past oscillating bluff-bodies (see forexample [60–64]).1.2.4 Types of thermoacoustic oscillationsPast studies show that, for the fixed combustor wall-temperature, the pressure signal featuresdifferent types. Specifically, the pressure signal may feature limit cycle [65–72], quasi-periodic[72–74], chaotic [72, 75–78], or intermittent [44, 79–84] type of oscillations. Occurrence of thesetypes of thermoacoustic oscillations depends on the fuel-air equivalence ratio. For example, forlean premixed swirl-stabilized flames, study of Kabiraj et al. [72] shows decreasing φ from 0.79to 0.78, and 0.7 causes the pressure signal to change from chaotic, to quasi-periodic, and thenlimit cycle oscillations, respectively. For very lean premixed flames, Ebi et al. [68] show whenφ decreases from 0.64 to 0.60 and 0.58, the pressure signal changes from featuring limit cycleto intermittent, and then chaotic oscillations, respectively. Usually, the Probability DensityFunction (PDF ), phase-space trajectory, Poincare´ map, and PSD are used for categorizing thedifferent types of the pressure oscillations with details discussed in the following.The PDF of the limit cycle pressure oscillations features a bimodal distribution [65, 67, 68],which peaks at ±amplitude of the oscillations. The PDF of the pressure oscillations amplitude isGaussian; the power spectrum density of the oscillations features a single peak; the phase-spacetrajectory (which is variation of the pressure oscillations versus the first and second derivativesof the pressure oscillations with respect to time) represents a three-dimensional ellipse [72, 85];and, the Poincare´ map (which is the intersection of the phase-space trajectory with the plane151.2. Backgroundof dp′/dt = 0) [86] features two separate clusters of points [72]. Compared to the limit cycleoscillations, the quasi-periodic oscillations are characterized by the frequent occurrence of large-amplitude oscillations followed by relatively small-amplitude oscillations [73, 85]. The PSD ofthe quasi-periodic oscillations usually features two or three dominant frequencies [72–74]. Also,the Poincare´ map of the quasi-periodic oscillations is a collection of points arranged in a torusshape [72, 85].Chaotic pressure oscillations commonly occur in large-scale gas turbine engines, see for ex-ample [87]. For chaotic oscillations, similar to the quasi-periodic oscillations, the pressure signalusually features two or three dominant modes. Casado [26], Alemela et al. [57], and Shahi etal. [2, 58, 88] replicated an industrial burner, and they showed that their combustor featuresthree dominant modes. Two of these modes occurred at frequencies of about 80 and 220 Hz,which nearly pertain to the quarter and three-quarter acoustic waves of their combustor. Thethird mode occurred at 600 Hz which was speculated to be linked to their combustor’s structuralnatural frequency [26]. Similar observations were reported in Bonciolini and Noiray [89], exceptthat the third mode (related to the structural natural frequency) was absent in their study.Compared to the limit cycle and quasi-periodic oscillations, results presented by Ebi et al. [68]indicate that the chaotic oscillations feature a Gaussian-shaped PDF . The Poincare´ map ofthe chaotic oscillations compared to those of limit cycle and quasi-periodic oscillations, featurea collection of scattered data points as shown by Kabiraj et al. [72].The intermittent oscillations are referred to the spontaneous switching of limit cycle andirregular chaotic oscillations [44, 79–81, 83, 84]. The intermittent oscillations have been observedin experimental setups related to atmospheric premixed flames, e.g. [44, 79], as well as elevated-pressure premixed and partially-premixed flames [18, 87, 90]. Since the intermittent pressure161.2. Backgroundsignal features both small- and large-amplitude oscillations, the PDF of p′ shows three dominantpeaks: one at about p′ ≈ 0 for very small-amplitude oscillations, and two peaks correspondingto the durations that the pressure oscillations feature limit cycle behavior [68]. Compared tothose of the limit cycle, quasi-periodic, and chaotic oscillations, the phase-space trajectory ofthe intermittent oscillations is a collection of small- and large-amplitude orbits [91].Several investigations [66, 89, 92–94] studied the behavior of the thermoacoustic oscillationsfor varying combustor wall-temperature. Studies of Kasthuri et al. [66], Lawn and Penelete[92], and Abduljalil et al. [93] show that during the transient combustor wall-temperature, thepressure signal may switch between large and small amplitude (referred to as noise in [66]) oscil-lations, which is referred to as the bursting oscillations in their investigations [66, 92, 93]. Duringthese oscillations, the phase-space trajectory features a large-amplitude orbit as well as a small-radius disk-shape trajectory positioned at the centre of the orbit [66]. Increasing the fuel-airequivalence ratio causes the amplitude of the oscillations to increase, while the large-amplitudeoscillations remain unchanged [66]. This combination of large- and small-amplitude oscillationsis referred to as the mixed-mode oscillations in Kasthuri et al. [66]. The Poincare´ map of suchoscillations is composed of four clusters of data points, two related to the large-amplitude os-cillations and two related to the small-amplitude oscillations [95]. Past investigations [92, 93]suggest after the combustor wall-temperature reaches a plateau, the pressure signal features alimit cycle behavior. Very few recent investigations [89, 94] have considered the influence oftransient combustor wall-temperature (and/or combustor thermal load) on pressure oscillationsamplitude. For example, studies of Bonciolini et al. [94] show that the chamber wall-temperaturedepends on their combustor thermal load. Changing this parameter leads to a periodic switchingbetween M- and V-shaped swirl-stabilized flames. This switching induces large-amplitude heat171.3. Thesis objectivesrelease rate and pressure oscillations. For a Bunsen burner, Mejia et al. [96] made similar obser-vations and showed that changing the burner rim temperature, changes the pressure oscillationsamplitude. Studies of [89, 94, 96] are of significant importance as they investigated the influ-ence of transient combustor wall- or burner rim-temperature on the thermoacoustic oscillationsamplitude. However, the effect of the transient combustor wall-temperature on the evolution ofthe oscillations characteristics (that is transitions between limit cycle, quasi-periodic, chaotic,etc.), and how the initial combustor wall-temperature influences these evolutions remain to beinvestigated.1.3 Thesis objectivesThe main goal of this study is to characterize thermoacoustic oscillations generated by apower generation unit. In several engineering applications, e.g. small-scale natural-gas-fuelledpower generators, reaching to the steady-state operation may take several minutes, during whichthe combustor wall-temperature increases in time affecting amplitude and type of thermoacousticoscillations. Also, in the operation of small-scale power generators, different types of thermoa-coustic modes may appear contributing positively or negatively to thermoacoustic oscillationsamplitude. However, related literature does not provide sufficient information addressing these.Thus, this study aims to address them through two objectives. First, it is aimed to address theeffect of transient wall-temperature on pressure oscillations characteristics. Second, the authoraims to study contributions of natural structural, acoustic, and intrinsic modes related to ourcombustor. Then, critical regions inside the combustor that are related to these modes are iden-tified. After this introduction, the experimental methodology is discussed in the second chapter.Results related to the first and second objectives are discussed in Chapters 3 and 4, respectively.181.3. Thesis objectivesConclusions are presented in Chapter 5.19Chapter 2Experimental MethodologyTwo versions of the power generator combustor are used in the present investigation. The firstversion of the combustor is a nearly identical replica of that used in the engineering application,in that it has a metallic wall and is not optically accessible. The second combustor is a simplifiedversion of the first combustor, but it is optically accessible. For both setups, air and fuel wereprovided from pressurized bottles. Methane grade 2 (99% chemical purity) was used as the fuel.SLA5850 and SLA5853 from Brooks were used to control the fuel and air volumetric flowrates,respectively. The accuracy of the fuel and air flowrates are 0.1 and 0.9 Standard Liters PerMinute (SLPM), which leads to a maximum uncertainty of about 4% for the tested fuel-airequivalence ratios. In the following, methodology related to the metallic and the opticallyaccessible combustors are provided.2.1 Experimental methodology related to the setup with themetallic combustorThe setup, measurement tools, and the tested experimental conditions along with the burnerstability map are described below.202.1. Experimental methodology related to the setup with the metallic combustor2.1.1 Experimental setupThe first experimental setup, i.e. the metallic combustor, is shown in Fig. 2.1(a). The setupis primarily composed of a combustion chamber and a preheating/premixing section, which areshown in Fig. 2.1(a) by yellow and pink dashed-rectangles, respectively. Figure 2.1(b) is anenlarged top view of the preheating/premixing section. This section includes two concentrictubes, whose schematics are shown in the solid black dashed-box of Fig. 2.1(c). The inner tube,which carries the hot combustion products, is shown by the blue dashed-lines in Fig. 2.1(c). Theouter tube carries the reactants (mixture of methane and air) in the direction opposite to that ofthe products, aiming to enhance heat transfer, decrease exhaust gas temperature, and increasesystem thermal efficiency. The exhaust stream shown in the figure is connected to a 1.3 mmetallic exhaust tube, a portion of which is shown in Fig. 2.1(a) by the red arrow. Both fueland air are completely mixed inside the preheating/premixing section. A Fenwal igniter was usedto ignite the mixture. A TDK-Lambda (model DPP120-24-1) power supply was used to operatethe igniter. Since the igniter supplied power can potentially influence the ignition dynamics[97–99], the supplied power was fixed at 120 W for all tested conditions. Fully-premixed flamesare stabilized on a metallic net, see Fig. 2.1(c). The combustion chamber is cooled using acurved water channel that circulates around the chamber, see Fig. 2.1(c). The results discussedin Chapter 3 are generated using this setup.2.1.2 DiagnosticsOne Setra 265 differential pressure transducer was used to measure the premixing/preheatingchamber pressure. The pressure transducer is capable of measuring up to about 6.2 kPa gauge,which is about twice of the maximum combustion chamber pressure. The pressure data is nearly212.1. Experimental methodology related to the setup with the metallic combustorFig. 2.1. (a) The first experimental setup, (b) top view of the preheating/premixing tube, and(c) schematic of the combustion chamber and the preheating/premixing tube.noise-free, and as a result, the pressure signals were not filtered for the results discussed inChapter 3. The pressure transducer port was installed 260 mm upstream of the injection plane,see Fig. 2.1(c). Due to the length of the combustion chamber and the preheating/premixingsection being relatively large (compared to the flame length), the flames can be considered tobe acoustically-compact. This means that, although the pressure at the flame region will bedifferent from that measured by the pressure transducer, it does not vary significantly acrossthe flame region. As a result, the spectral content of the pressure signal can be used to describethe pressure oscillations at the flame region. Although the pressure oscillations amplitude at222.1. Experimental methodology related to the setup with the metallic combustorthe flame region will be different from that measured by the pressure transducer, the trendsassociated with the effect of the governing parameters on the pressure oscillations amplitudecan be deduced from the pressure transducer signal.Three AD8497 K-type thermocouples from Analog Devices were used to measure the com-bustor wall (Tw), warm reactants (Tr), and the exhaust gas (Te) temperature. The plane thatcontains the thermocouple used to measure the combustor wall-temperature, Tw, is positioned55 mm downstream of the injection plane; Tr is measured at the injection plane; and, Te ismeasured at the end of the metallic exhaust tube, as shown in Fig. 2.1(c). The precision of thethermocouple measurements is about 2-6 K, which is provided by the producer. Both the pres-sure transducer and the thermocouples data were recorded simultaneously using a LabJack T7series at 10 kHz by a DAQ board which is schematically shown in Fig. 2.1(c). The LabJack datarecorder has a 16-bit high speed analogue to digital converter whose generated sampling rate canbe distributed between maximum of 14 analogue inputs. Three of these analogue inputs wereused for the measurements. A tertiary magnetic-inductive flowmeter from IFM (model 6004)was used to measure the water flowrate of the combustion chamber cooling system. This massflow controller, and air and fuel mass flow controllers operate separately and are not controlledby the LabJack.2.1.3 The tested conditions and the stability mapIn total, three parameters, the fuel-air equivalence ratio (φ), the air volumetric flowrate(Q˙air), and the initial combustor wall-temperature (Tw,i) were changed in the experiments relatedto the setup shown in Fig. 2.1 independently. Here φ was varied between 0.7 and 1.4, and Q˙air waschanged between 50 and 130 SLPM. The resolution of varying φ and Q˙air were 0.05 and 5 SLPM,respectively. The Tw,i was either set to 290 or 680 K. The former and the latter conditions232.1. Experimental methodology related to the setup with the metallic combustorare referred to as the cold- and warm-start conditions. In total, 9 conditions denoted by thealphabetic symbols were tested. Details related to these conditions (the fuel and air flowrate,φ, the mean bulk flow velocity, and the initial combustor wall-temperature) are summarized inTable 2.1. The warm-start conditions are highlighted by the asterisk sign for clarity purposes.Table 2.1. Tested experimental conditions. The asterisk sign refers to warm-start condition.Tw,i is the initial combustor wall-temperature.Condition A∗ B C D B–D E∗ F G HQ˙CH4 (SLPM) 6.0 7.2 8.8 10.4 7.2–10.4 3.8 3.7 5.3 7.4Q˙air (SLPM) 90 90 110 130 90–130 50 55 60 70φ 0.75 0.9 0.9 0.9 0.9 0.85 0.75 1 1.2U (m/s) 2.3 2.3 2.8 3.3 2.3–3.3 1.3 1.4 1.5 1.8Tw,i (K) 680 290 290 290 290 680 290 290 290The stability map, that is the combination of experimental conditions for which the fuel-airmixture ignites and the flame is stable for the duration of the experiments, was developed. Thismap is shown in Fig. 2.2. The blue cells in the map pertain to the conditions for which the flamewas stable and did not blow-out or flash-back. The yellow cells pertain to conditions that themixture was ignited; however, the flame was not stable, and it extinguished after a few seconds.The pink cells pertain to conditions that the mixture was not ignited initially; however, if theburner started from a condition related to a blue-coloured cell first, and then φ and Q˙air werechanged to those of the pink cell, the flame remained stable. For the conditions related to thered cells, the mixture was not ignited, and it was not possible to stabilize the flames even if theburner was initially started from a condition related to the blue cells.For all tested conditions, except the condition that pertains to transitioning from B to D,referred to as B−D, the fuel-air equivalence ratio and the air flowrate were kept constant duringthe experiments. For condition B − D, however, only the fuel-air equivalence ratio was keptconstant, and the air flowrate was increased from 90 to 130 SLPM. For the B−D condition, the242.1. Experimental methodology related to the setup with the metallic combustorFig. 2.2. Stability map of the power generator shown in Fig. 2.1. Cells with blue colour pertainto conditions that the mixture was ignited and the flame was stable. For the conditions with theyellow colour, the mixture was ignited, but it was extinguished after a few seconds. Conditionswith pink colour pertain to those that flame could only be stabilized if the burner was warmed-up initially. The mixture cannot be ignited even if the burner was warmed-up for conditionsdenoted by the red colour. Note that conditions with the asterisk sign are associated with thewarm-start, i.e. Tw,i = 680 K. The conditions without the asterisk refer to Tw,i = 290 K.flame was ignited for φ and Q˙air related to condition B, and the burner operated for about 135 s.Then, the fuel and air flowrates were increased during 4 steps that were separated by 5 s untilthe condition D was reached. Note that, during the entire transition from B to D condition,the fuel-air equivalence ratio is constant and equal to 0.9. The air and fuel flowrates variation252.2. Experimental methodology related to the optically accessible setuprelated to condition B −D is shown in Figs. 2.3(a) and (b), respectively. The steps related tothe transition between B and D conditions are also highlighted in the insets of the figures.010020005100 200 400 600 800 1000 12000500100005125 145 16570140125 145 165681012Fig. 2.3. Variations of the fuel (a) and air (b) flowrates for condition B − D. The fuel-airequivalence ratio is kept constant and equal to 0.9 throughout the experiment.2.2 Experimental methodology related to the opticallyaccessible setupThe second experimental setup, diagnostics, and the tested conditions are detailed below.2.2.1 Experimental setupThe setup shown in Fig. 2.1 was redesigned by the power generator producer providingoptical accessibility. This setup is similar to that used in [100]. The technical drawing andthe schematic of the optically accessible setup are shown in Figs. 2.4(a) and (b), respectively.This setup is composed of a plenum, an optically accessible main tube (the light blue regionin Figs. 2.4(a) and (b)), six optically accessible exhaust tubes (the red regions in Figs. 2.4(a)and, (b)), a stainless steel stagnation wall that holds the main and exhaust tubes together andreverses the flow direction, a torus-shaped flame-holder (with small and large diameters of 1.5262.2. Experimental methodology related to the optically accessible setupand 24 mm), and a turbulence generator. After ignition, the flames are stabilized on the flame-holder shown in Figs. 2.4(b) and (c). The utilized turbulence generator generally helps increasingthe burning rate and decreasing the flame height. Then, the combustion products are directedinto the small exhaust tubes with inner diameter of 10.2 mm. All exhaust tubes discharge intothe same manifold, which directs the combustion products into a 1.3-meter-long metallic exhausthose through which the exhaust gas cools down due to the condensation and then it is guidedinto a naturally ventilated exhaust. A Cartesian coordinate system was positioned at the centerof the flame-holder. The x−axis is positioned along the main tube centerline, and the y−axisis along the radial direction and normal to the ground. The z−axis is normal to both x− andy−axes.7RSUHVVXUHWUDQVGXFHU([KDXVWPDQLIROG$FFHOHURPHWHU2SWLFDOO\DFFHVVLEOHH[KDVXWWXEHV2SWLFDOO\DFFHVVLEOHPDLQWXEH0HWKDQHDLU([KDXVWPHWDOOLF6WDJQDWLRQZDOOSURGXFWV&RPEXVWLRQPP,JQLWHUFHUDPLFLQVXODWRU ,JQLWHUWLS PPGLDPHWHU 3HUIRUDWHGSODWH)ODPHKROGHU0DLQWXEHIORZ0HWKDQHDLU6WDJQDWLRQSODWHORFDWLRQ F 1R]]OH PPGLDPHWHU PPPP PP (a) (b)(c) (d),JQLWHUKRVHPL[WXUHFig. 2.4. (a and b), (c), and (d) are the technical drawings and schematic of the experimentalsetup, detailed drawing of the burner, and the utilized turbulence generator, respectively.272.2. Experimental methodology related to the optically accessible setupThe main tube inner diameter is 33.0 mm, with the technical drawing shown in Fig. 2.4(c).The Fenwal igniter used for the first setup (discussed in section 2.1.1) was also used here. Theigniter rod was positioned along the main tube centerline. The rod tip diameter is 1.7 mm, andit was bent 90◦ as shown in Figs. 2.4(b and c). The igniter rod tip location with respect tothe flame-holder, S, was changed depending on the tested experimental conditions. The TDK-Lambda power supply used for the first setup was also used here. The power supply dischargedpower was fixed at 120 W similar to the experiments performed for the first setup. After ignition,the flames are stabilized on the flame-holder shown in Figs. 2.4(b) and (c).The technical drawing of the turbulence generator is shown in Fig. 2.4(d). The generatorfeatures hexagonally arranged holes with a diameter of 1.7 mm generating a blockage ratio ofnearly 70%. The central hole in the turbulence generator is to allow for installation of the igniterrod. The generator shown in Fig. 2.4(d) was used for all tested conditions and was positioned45 mm upstream of the flame-holder. This is about 1.4 times the burner inner diameter, similarto [101–104]. The turbulence generator that is utilized upstream of the main tube flame issimilar to that used in [105].2.2.2 DiagnosticsSynchronized and high-speed pressure and OH∗ chemiluminescence measurements were per-formed. The Setra 265 differential pressure transducer was connected to a pressure tap located115.0 mm upstream of the flame-holder using a 642.1 mm-long wave-guide. This leads to atime lag in the the pressure data collection. This lag was estimated (2.2 ms), and the pressuredata was corrected accordingly. The output of the pressure transducer is in volts (maximum5 V), and the pressure values were calibrated using the producer calibration instructions. Theresolution of the pressure transducer signal is 0.8 mV/Pa.282.2. Experimental methodology related to the optically accessible setupThe chemiluminescence imaging system consists of a high-speed camera (Photron FastcamNova S12) as well as an image intensifier (Invisible vision UVi). The intensifier was equippedwith a UV Nikon lens, with focal length of 105 mm. The lens was equipped with an OH∗ band-pass filter (center wavelength of 310 nm and bandpass width of 20 nm). The camera exposuretime and the intensifier gate were 25 and 20 μs, respectively. The intensifier gain was set to 38%for the tested conditions, depending on the field of view. The camera output signal was used tosynchronize the pressure measurements (at acquisition frequency of 50 kHz by LabJack T7 datalogger) and the image intensifier at 5 kHz. The camera was positioned along the z−axis. Thechemiluminescence signal has contributions from both OH∗ and CO∗2 chemiluminescence near310 nm [106]. However, the bandpass filtered signal at 310 nm has been considered to be relatedto heat release rate of fully premixed flames [107–109], and the OH∗ signal is utilized here forstudying the dynamics of the heat release rate.The vibrations of the exhaust metallic hose were recorded using a 3DM-GX3 attitude head-ing reference system. This sensor is attached to the exhaust hose as shown in Fig. 2.4(a). Thesevibration measurements were recorded when the system was turned off and were used to estimatethe system structural frequency. A mechanical hammer was used to excite the system. The vi-brometer facilitates the triaxial acceleration measurements. The maximum range of accelerationis +/- 5g with 0.01 % accuracy. The accelerations along x, y, and z directions are recorded ata sampling rate of 500 Hz (maximum possible sampling rate) and then the acceleration signalis integrated twice to estimate the displacement along x, y, and z directions versus time. Sincethe setup is initially at rest, the initial values of velocities and displacements are set to zero forthe integrations, with details discussed in Chapter 4.292.2. Experimental methodology related to the optically accessible setup200 400 6001120240200 400 6000 40 80 85-150150 40 80 85Fig. 2.5. (a) Raw, (b) background subtracted and sensitivity corrected (c) binned, and (d)median-based filtered images.2.2.3 Data reductionSince the pressure and displacement signals are nearly noise-free, they were directly usedin the analysis. The OH∗ chemiluminescence signal requires data reduction which is discussedhere. A series of processing steps were applied to the OH∗ chemiluminescence raw images beforeanalysis. The processing steps are demonstrated in Fig. 2.5. Raw chemiluminescence data isshown in Fig. 2.5(a). The raw images were first subtracted by a mean background image, whichwas acquired from 1000 images with the UV lens blinded. Then, the background subtractedimages were corrected by the camera and intensifier sensitivity array, with the correspondingresults presented in Fig. 2.5(b). The sensitivity array was obtained by, first, background sub-tracting, and then averaging the images taken from a FotodioX (C-300RS FlapJack) LED lamp,which was positioned against the camera lens. The lamp operated at 20% brightness, and theintensifier gain and gate were set to 38% and 20 μs, matching those of the tested experimentalconditions. The intensifier sensitivity array was maximized near the lens centre and decreased302.2. Experimental methodology related to the optically accessible setupto 20% of the maximum towards the sensor edges. It was confirmed that the brightness% andposition of the LED lamp relative to the intensifier lens as well as the tested intensifier gainand gate do not change the sensitivity array significantly. The sensitivity corrected images werebinned by 4 × 4 pixels2 windows, and the signal inside each bin was averaged and utilized foranalysis. This was performed to reduce the post-processing time and increase the signal-to-noise ratio [12, 18, 110, 111]. The binned data corresponding to results shown in Fig. 2.5(b)is presented in Fig. 2.5(c). To reduce the noise, a 4 × 4 median-based algorithm in MATLABwas used to filter the data presented in the Fig. 2.5(c), with the corresponding results shown inFig. 2.5(d). The processed images after the median-based filtering were used for analysis.2.2.4 Tested conditions and the flame configurationsThe tested conditions are presented in Tables 2.2 and 2.3 related to the fuel-air equivalenceratios of 0.7 and 0.9, respectively. These fuel-air equivalence ratios were tested as per therecommendation of the industry partner. The igniter rod tip location, and the air volumetricflowrate were changed in the experiments. The former was set to either −5, 1.5, 2.5, or 5 mm;the latter was varied between 120 and 180 standard liters per minute (SLPM) with incrementsof 20 SLPM. In total, 32 experimental conditions were tested. The Reynolds numbers listed inthe tables were estimated based on the diameter of the nozzle (33 mm) and air/fuel mixtureviscosity estimated at room temperature.The type of the generated flame (Bunsen or M-shaped) for each tested experimental con-dition is shown in Figs. 2.6(a) and (b) related to the fuel-air equivalence ratio of 0.7, and 0.9,respectively. The indices 1 to 4 highlight the relevant location of the igniter, which is alsoshown on the vertical axis of Fig. 2.6. Depending on the igniter rod location and the tested airflowrate, the flames were either stabilized on the flame-holder (generating Bunsen flames), or on312.2. Experimental methodology related to the optically accessible setupTable 2.2. Tested conditions with the fuel-air equivalence ratio set to 0.7 . The unit of Q˙ is inStandard Liters Per Minute.Flame Q˙air Q˙CH4 S (mm) ReA1 120 9.14 -5.0 5158A2 120 9.14 1.5 5158A3 120 9.14 2.5 5158A4 120 9.14 5.0 5158B1 140 10.66 -5.0 6018B2 140 10.66 1.5 6018B3 140 10.66 2.5 6018B4 140 10.66 5.0 6018C1 160 12.18 -5.0 6878C2 160 12.18 1.5 6878C3 160 12.18 2.5 6878C4 160 12.18 5.0 6878D1 180 13.71 -5.0 7738D2 180 13.71 1.5 7738D3 180 13.71 2.5 7738D4 180 13.71 5.0 7738Table 2.3. Tested conditions with the fuel-air equivalence ratio set to 0.9 . The unit of Q˙ is inStandard Liters Per Minute.Flame Q˙air Q˙CH4 S (mm) ReE1 120 11.75 -5.0 5263E2 120 11.75 1.5 5263E3 120 11.75 2.5 5263E4 120 11.75 5.0 5263F1 140 13.71 -5.0 6140F2 140 13.71 1.5 6140F3 140 13.71 2.5 6140F4 140 13.71 5.0 6140G1 160 15.67 -5.0 7017G2 160 15.67 1.5 7017G3 160 15.67 2.5 7017G4 160 15.67 5.0 7017H1 180 17.63 -5.0 7894H2 180 17.63 1.5 7894H3 180 17.63 2.5 7894H4 180 17.63 5.0 7894322.2. Experimental methodology related to the optically accessible setupboth the flame-holder and the igniter rod (generating M-shaped flames). In Fig. 2.6, the Bunsenand M-shaped flames are distinguished by the white and pink background colours, respectively.Flame D2 condition is highlighted by both background colours. This is because the D2 conditionfeatures unstable flame shape meaning that it can feature both Bunsen and M-shaped flamesthrough a flame shape transition. Specifically, D2 mostly features Bunsen flames. However,sometimes the Bunsen flames transition to M-shaped flames several seconds after the ignition.This transition is discussed in detail in Appendix A.120 140 160 180-51.52.55120 140 160 180Fig. 2.6. The alphabets in (a), and (b) pertain to the tested conditions presented in Tables 2.2and 2.3, for φ = 0.7 and 0.9, respectively. White and pink background colours correspond toBunsen and M-shaped flames, respectively. D2 experimental condition can feature both Bunsenand M-shaped flames.Figure 2.7(a) and (b) present time-averaged OH∗ chemiluminescence related to Flames D1and D2, pertaining to Bunsen and M-shaped flames, respectively. Representative time-averagedOH∗ contours are highlighted in Figs. 2.7(a) and (b) by the black solid curves, showing thegeneric topology of the tested flames, and the reason of the M-shaped flame terminology (seethe rotated “M” in Fig. 2.7(b)). Also overlaid on the figures are black dashed-ellipses highlightingthe exterior wings of the flame stabilized on the torus-shaped flame-holder. The white dashed-region in Fig. 2.7(b) highlights the flame stabilized on the igniter rod.332.2. Experimental methodology related to the optically accessible setup0 25 50 75 85-15015-15015Fig. 2.7. (a) and (b) are the time-averaged OH∗ chemiluminescence images related to Flames D1and D2, pertaining to Bunsen and M-shaped flames respectively. The dashed-ellipses highlightthe exterior wings of the flames formed on the torus-shaped flame-holder. The white dashed-region in (b) highlights the flame formed on the igniter rod.In the following, Chapter 3 summarizes the findings of this thesis related to the first ex-perimental setup. These findings aim to address the first objective of the thesis. Results inChapter 4 correspond to the optically accessible setup. Chapter 4 aims at understanding thecontributions of different thermoacoustic modes to the Rayleigh gain, which is related to thesecond objective. The transition between the Bunsen and M-shaped flames for condition D2 isdiscussed in Appendix A.34Chapter 3The effect of wall-temperature onthermoacoustic oscillationsIn this chapter, the first experimental setup with details discussed in section 2.1.1 is used. Theevolution of the thermoacoustic oscillations while the wall-temperature increases both for cold-and warm-start conditions are investigated. Then, the relations between St and He numberswith Re number are studied to identify the mechanism driving the thermoacoustic oscillations.3.1 Pressure oscillations behaviorThe time history of the pressure oscillations pertaining to conditions A∗, B, C, and B −Dare shown in Figs. 3.1(a-c), (d), (e), and (f), respectively. The rest of the tested conditionsfeatured similar behavior and are not presented here for brevity. The black and blue data inFig. 3.1 are the pressure oscillations and the pressure oscillations amplitude (p˜′), respectively.The Hilbert transform [112] was used to estimate the instantaneous p˜′, that isp˜′(t) = |H [p′(t)] |, (3.1a)H [p′(t)] = 1π∫ +∞−∞p′(τin)τin − tdτin, (3.1b)353.1. Pressure oscillations behavior100 200 400 500 600-20280 82 84 86 88 90-202560 562 564 566 568 57039 40 41 42 43 44 45 46 47-202280.6 280.65 280.7 280.75 280.8 280.85 280.9 280.95 281-202819 819.5 820 820.5 821 821.5 822-202Fig. 3.1. (a) Variation of the pressure oscillations for condition A∗. (b) and (c) are insets of(a). (d), (e), and (f) are samples of the pressure oscillations pertaining to conditions B, C∗, andB–D, respectively. The blue curves are the instantaneous pressure oscillations amplitude.where t is time, and τin is an integration variable. As can be seen in Fig. 3.1(a), the pressureoscillations pertaining to condition A∗ feature several large-amplitude oscillations that are sep-arated by small-amplitude oscillations. Two insets related to the pressure oscillations data forcondition A∗ are illustrated in Figs. 3.1(b) and (c), demonstrating transitions between small-and large-amplitude oscillations. Increasing the fuel-air equivalence ratio from 0.75 to 0.9 (con-dition B) leads to occurrence of chaotic events. A representative event related to these chaoticoscillations for condition B is demonstrated in Fig. 3.1(d). For condition C, both the transitionsrelated to condition A∗ as well as the limit cycle oscillations are observed, with an example ofthe latter shown in Fig. 3.1(e). Transitioning from condition B to D leads to occurrence of363.1. Pressure oscillations behaviorbursting oscillations (combination of large-amplitude and noise-level oscillations) in addition tothe previous types of oscillations shown in Figs. 3.1(a-e).The results presented in Fig. 3.1 show that the pressure oscillations amplitude feature atime-dependent behavior for all tested conditions. This behavior has been previously observedand reported in the studies of [66–78, 92, 93]. However, in these investigation, only one type ofthe oscillations, that is either limit cycle [65–72], quasi-periodic [72–74], bursting [66, 92, 93],mixed-mode [66], and chaotic [72, 75–78] are observed at a fixed fuel-air equivalence ratio,mean bulk flow velocity, and combustor wall-temperature. Compared to past investigations,in the present study, the combustor wall-temperature changes, and as a result, combinationsof different pressure oscillations types have been observed at a given fuel-air equivalence ratioand mean bulk flow velocity. The discussions presented in this chapter aim to characterize thetransitions between different types of the pressure oscillations during both cold- and warm-startoperation of the system when the combustor wall-temperature increases with time. The rest ofthe results are grouped into four subsections. In subsection 3.2, the probability density functionas well as the power spectrum density of the long time-period pressure oscillations are presentedand discussed. The long time-period analysis allows to investigate the general characteristicsof the pressure oscillations, e.g. the dominant modes and their dependence on φ and U . Insubsections 3.3 and 3.4, the dynamics of the pressure oscillations for small (290 K) and large(680 K) combustor wall-temperature transitioning to a steady state value (∼850 or 890 K,depending on the tested condition) have been discussed, respectively. In the last subsection, therelations between the Helmholtz and Strouhal numbers and the Reynolds number during thetransient operation of the system are discussed.373.2. Long time-period characteristics of the pressure oscillations3.2 Long time-period characteristics of the pressureoscillationsPower spectral analysis of the pressure oscillations was performed to investigate the apparentdominant modes as well as the influences of the fuel-air equivalence ratio and the mean bulk flowvelocity on these modes. Also, the normalized Probability Density Functions (PDF ∗) of thepressure oscillations and their amplitude are obtained and discussed here for long time-periodoscillations. The results presented in this subsection are used as a benchmark for comparisonswith those obtained from the time-dependent analysis, which are discussed in the followingsubsections.Power spectrum densities of the pressure oscillations for conditions A∗, B, C, and B − Dare shown in Fig. 3.2. These PSDs are calculated based on the pressure signal collected for atleast 10 minutes (6 million data points for each condition). The results are normalized by thearea under the corresponding PSD and are stepped for clarity. For comparison purposes, thepower spectrum density of the pressure oscillations for the air and fuel flowrates pertaining tocondition C, but without the flame, are also presented. As can be seen, for the non-reactingflow condition, no discernible peak was present in the PSD of the pressure oscillations. Forreacting conditions, however, the results suggest that the pressure oscillations generally featurethree dominant frequencies (f1, f2, and f3) which are dependent on the total volumetric flowrate(or U) and the fuel-air equivalence ratio. The results in Fig. 3.2 show that changing the formerdoes not change f1−3 significantly; however, influence of the fuel-air equivalence ratio on thesefrequencies is significant (compare results related to conditions A∗ and B). Additionally, thepower spectrum densities show existence of two peaks pertaining to small frequencies of aboutf2 − f1 = 10 and f3 − f2 = 35 Hz. The peaks related to the linear combination of the dominant383.2. Long time-period characteristics of the pressure oscillationsmodes frequencies have also been observed and reported in past investigations, see for example[32, 72].200 300 400 50000.00050.0010.00150.0020.00250.0030.00350.0040.0045Fig. 3.2. Representative normalized power spectrum density of the pressure oscillations.The normalized probability density functions of the pressure oscillations and their amplitudeare presented in Figs. 3.3(a) and (b), respectively. The normalization of the PDF s in Fig. 3.3is performed with respect to the total number of the pressure data for the corresponding testedconditions. For clarity of presentation purposes, the PDF ∗ values are stepped. For conditionA∗, the probability density function of the pressure data features four peaks; and the PDF ofthe pressure oscillations amplitude features a bimodal distribution. The pressure oscillations393.2. Long time-period characteristics of the pressure oscillationsamplitudes related to these modes are highlighted by p˜′1 and p˜′2 in Fig. 3.3(b). The transitionsbetween large- and small-amplitude pressure oscillations shown in Figs. 3.1(a-c) lead to the bi-modality of p˜′ probability density function, which is analogous to existence of four peaks in thepressure oscillations PDF , as evident in Fig. 3.3(a).0 0.5 1 1.5 200.0050.010.0150.020.0250.03-2 -1 0 1 200.0020.0040.0060.0080.0100.012Fig. 3.3. Representative normalized probability density functions of (a) the pressure oscillationsand (b) the pressure oscillations amplitude.Increasing the fuel-air equivalence ratio from 0.75 to 0.9 (i.e. changing from condition A∗to B) decreases p˜′1 and p˜′2 as shown in Fig. 3.3(b). Note these small-amplitude oscillationsare not shown in Fig. 3.1(d). Here, p˜′2 is related to events during which the pressure signalfeatures limit cycle oscillations. Comparison of the time histories of the oscillations shown inFig. 3.1(d) with those in Figs. 3.1(b and c) suggests the transitions between small- and large-amplitude oscillations are relatively more often and random for condition B. In fact, this leadsto the large probabilities for p′ values that are close to zero and the nearly top-hat PDF of p′ in403.3. Transient behavior of the pressure oscillations for cold-start conditionsFig. 3.3(a) for condition B. At φ = 0.9, increasing the air flowrate from 90 to 110 SLPM leads toemergence of large-amplitude limit cycle oscillations as demonstrated in Fig. 3.1(e). This is alsoevidenced by the relatively large value of the pressure oscillations amplitude PDF at p˜′2. Thelimit cycle oscillations are also evidenced by the nearly bimodal profile of p′ shown in Fig. 3.3(a).The pressure signal related to condition B −D features limit cycle, quasi-periodic, chaotic, andbursting behavior. As shown in Fig. 3.1(f), the time period associated with occurrence of small-amplitude oscillations (related to the bursting behavior) is relatively large, leading to large PDFvalues at p′ ≈ 0 in Fig. 3.3(a).Although the PSD and PDF of the pressure signals calculated for long time-period oscilla-tions allow for studying the general effect of the fuel-air equivalence ratio and the mean bulk flowvelocity on the oscillations amplitude and the dominant modes, the long time-period analysissmears out the effect of the transient combustor wall-temperature on the oscillations. Duringthe transient operation of the combustor, the system dynamics transitions between several typesof the oscillations and these can be best captured by the PSD and the PDF analysis performedfor short time-period sliding windows. In the following two subsections, the system dynamicsfor the transient combustor wall-temperature related to both cold- and warm-start conditionsare discussed.3.3 Transient behavior of the pressure oscillations forcold-start conditionsThe time dependent characteristics of the pressure oscillations for all tested cold-start condi-tions are similar; thus, only representative results related to one condition (B −D) is presentedand discussed. The solid parallel lines in Figs. 3.4(a) and (b) pertain to the time period during413.3. Transient behavior of the pressure oscillations for cold-start conditionswhich the mean bulk flow velocity increases from 2.3 m/s (related to B) to 3.3 m/s (related toD) while φ is kept constant at 0.9. The measured combustor wall-temperature for conditionB −D is shown by the solid black curve in Fig. 3.4(a). The results in the figure show that thecombustor wall-temperature initially increases and then plateaus. The following formulation isproposed to mimic this time-dependent behavior of the combustor wall-temperature:T − T0T∞ − T0 =2πarctan(tt0− 1). (3.2)In Eq. (3.2), T0 is the temperature at the time instant that the mean bulk flow velocity pertainsto that of condition D. This time is referred to as t0 and is highlighted in Fig. 3.4(a). Thevalues of Tw estimated from Eq. (3.2) are shown by the black dotted-curve in Fig. 3.4(a). Ascan be seen, the measured temperature and the estimated values from the right-hand-side ofEq. (3.2) are close. The above formulation is later used to study the time dependent behaviorof the pressure oscillations frequency in this subsection.The spectrogram of the pressure oscillations pertaining to the B − D condition is shownin Fig. 3.4(b). During the time period between the ignition, t = 29 s, and t0 = 153 s, thepressure oscillations feature two modes with relatively small frequencies. See Fig. 3.4(b). Fort0  t  450 s, the pressure oscillations feature one dominant frequency that increases abruptlyat about t = 200 s and then gradually increases with time. For 450 s  t  700 s, the threemodes of the pressure oscillations (with f1−3 frequencies) shown in Fig. 3.2 appear. Then, aftert ≈ 700 s, the first mode of the oscillations diminishes and the second and third modes (with f2and f3 frequencies) continue to co-exist. Additionally, the linear combination of the dominantfrequencies (f3−f2) appear in the spectrogram of the oscillations, which is also observed in pastinvestigations [32, 72]. The results in Fig. 3.4(b) show that for 1000 s  t  1350 s, f2 and f3423.3. Transient behavior of the pressure oscillations for cold-start conditionsFig. 3.4. (a) Combustor wall-temperature variation. The black solid and dotted curves arethe experimentally measured temperature and the right-hand-side of Eq. (3.2), respectively.(b) Spectrogram of the pressure oscillations. The dotted curves in (b) are the right-hand-sideof Eq. (3.4) for the three oscillations modes. (c) Variation of the pressure oscillations PDFestimated for a sliding time window. (d-g), (h-k), and (l-o) are the pressure time histories,phase-space trajectories, and the Poincare´ maps, respectively. The results in (d, h, l), (e, i,m), (f, j, n), and (g, k, o) pertain to t = [29 31], [302 304], [847.4 848.6], and [1482 1486] s,respectively. The results pertain to condition B −D (a cold-start condition).433.3. Transient behavior of the pressure oscillations for cold-start conditionsplateau; and, after t ≈ 1350 s, the second mode diminishes and only the third mode exists.Using the ideal gas equation of state, it can be shown that the frequencies of dominant modesshould be proportional to the square root of the gas temperature for acoustically-dominatedsystems. This means that, for the ith mode, the dominant frequency of the oscillations is givenbyfif∞, i=√TT∞, (3.3)with f∞,i being the dominant frequency of the ith mode when T = T∞. Combining Eqs. (3.2) and (3.3),it is shown thatfi = f∞,i√T0/T∞ + (2/π)(1− T0/T∞) arctan(t/t0 − 1). (3.4)The right-hand-side of Eq. (3.4) is estimated, and the corresponding results are shown by thesolid dashed-curves in Fig. 3.4(b) for all three modes of the oscillations. As can be seen, the val-ues of fi estimated from the experiments and Eq. (3.4) nearly match, suggesting that Eq. (3.4)is suitable for estimating the time-dependent behavior of the pressure oscillations frequency.Equation (3.4), however, cannot predict the transition between the modes of the pressure os-cillations. The transition between the pressure oscillations modes is speculated to be linked topotential changes in the flame shape inside the combustion chamber, as observed in past studies[30, 79, 83, 94, 113]. Nevertheless, the underlying reason for these transitions is not discussedin this chapter. Instead, the characteristics of the oscillations during the transient combustorwall-temperature are investigated in the following.The PDF of the pressure oscillations was estimated for a sliding time window with a lengthof 3.3 s (33000 pressure data points). The corresponding results are presented in Fig. 3.4(c).443.3. Transient behavior of the pressure oscillations for cold-start conditionsThe results in the figure show that the pressure oscillations transition between the Gaussian-type and bi-modal PDF s. In order to investigate the system dynamics during such transitions,the phase-space trajectories and the Poincare´ maps were utilized. The phase-space trajectorieswere obtained by presenting the variations of the pressure oscillations along with their firstand second derivatives with respect to time. The Poincare´ map is the collection of pointspertaining to the intersection of an arbitrary plane and the phase-space trajectories. In thepresent study, the plane of dp′/dt = 0 was used as the Poincare´ section. Note that in thepast investigations, see for example [66, 72, 73, 85, 86], the phase-space trajectories are usuallypresented as (p′(t), p′(t+τ), p′(t+2τ)), with τ being a time-lag set to estimate the Poincare´ maps.However, presenting the phase-space trajectories using the pressure oscillations derivatives (themethod used here) leads to the similar results [114].The pressure oscillations, the corresponding phase-space trajectories, and the Poincare´ mapsrelated to four time windows, highlighting the different characteristics of the pressure oscilla-tions, were selected. These time windows pertain to t = [29 31], [302 304], [847.4 848.6], and[1482 1486] s. These windows are shown by the dashed-rectangles in Fig. 3.4(c). For clarityof discussions, the time instants related to either beginning or centres of these windows areshown by t = 29, 303, 848, and 1484 s on the time axis of Fig. 3.4(c). The pressure oscillations,the phase-space trajectories, and the corresponding Poincare´ maps related to these four timewindows are presented in Figs. 3.4(d-g), (h-k), and (l-o), respectively. Note that the blue circu-lar and red cross data symbols in Figs. 3.4(l-o) pertain to Poincare´ sections with positive andnegative d2p′/dt2, respectively.The results in Fig. 3.4 show that after the ignition (at t = 29 s) and over approximately a270 s period, the combustor wall-temperature increases from T = 290 to 700 K. During this453.3. Transient behavior of the pressure oscillations for cold-start conditionstime period, the pressure oscillations feature a Gaussian-type PDF as can be seen in Fig. 3.4(c).Both quasi-periodic and chaotic oscillations probability density functions may feature Gaussian-type distributions. Results presented in Kabiraj et al. [72] show that the Poincare´ map ofthe oscillations can allow for distinguishing the type of the oscillations. They show that thechaotic oscillations feature two scattered clusters of points; however, that of the quasi-periodicoscillations feature two hollow toroidal structures. The results presented in Fig. 3.4(l) show twoscattered clusters of points, and as a result, the pressure oscillations feature chaotic behaviorduring t = [0 300] s.During t ≈ [300 450] s, the pressure oscillations feature a bi-modal PDF , and the spectro-gram of the oscillations shows a single dominant frequency, as shown in Figs. 3.4(b) and (c).The phase-space trajectories and the Poincare´ maps feature orbital and two separate clustersof points, as shown in Figs. 3.4(i) and (m), respectively. Such limit cycle pressure oscillationswith the Poincare´ maps and the phase-space trajectories have also been observed and reportedin past studies. See for example [72].During t ≈ [450 700] s, the PDF of the pressure oscillations becomes Gaussian, and thePoincare´ maps feature two clusters of points. These oscillations are chaotic and feature charac-teristics similar to those shown in Figs. 3.4(d, h, and l). For brevity purposes, the results relatedto t ≈ [450 800] s are not presented. At t ≈ 848 s, the pressure signal features sudden transitionto large-amplitude limit cycle oscillations followed by noise level oscillations. This behavior isdemonstrated in the pressure oscillations and their PDF shown in Figs. 3.4(c) and (f). ThePoincare´ maps feature two clusters of points that are initially well-separated, and they movetowards the origin as the pressure signal transitions from limit cycle to noise-level oscillations.This is shown by the arrow in Fig. 3.4(n). The results shown in Figs. 3.4(j) and (n) suggest the463.4. Transient behavior of the pressure oscillations for warm-start conditionspressure signal features bursting oscillations which is similar to those observed and reported in[66, 92, 93].The results in Fig. 3.4(c) show that, after t ≈ 848 s, the PDF of the pressure oscillationsremains Gaussian up to t ≈ 900 s. However, with the plateau of the combustor wall-temperature,the PDF of the pressure oscillations transitions to a bi-modal distribution. See Fig. 3.4(c). ThePoincare´ maps of the pressure oscillations feature two well-separated clusters of points, as shownin Fig. 3.4(o). Comparison of the results pertaining to t = [302 304] s and t = [1482 1486] ssuggests that the limit cycle oscillations may occur both when the combustor wall-temperatureis increasing as well as when the temperature plateaus. However, the amplitude of these limitcycle oscillations is smaller at the larger combustor wall-temperature.3.4 Transient behavior of the pressure oscillations forwarm-start conditionsThe results presented in the previous subsection suggest that, for the cold-start conditions,the system features limit cycle oscillations when the combustor wall-temperature plateaus. How-ever, the influence of the warm-start condition on the characteristics of the pressure oscillationswere not studied, and these are investigated here. This is important, since the correspondingresults along with those from the previous subsection will allow the devising of guidelines thatfacilitate improved operation of the power generation system in terms of the thermoacousticoscillations.Similar pressure oscillations characteristics related to both warm-start tested conditions (A∗and E∗) were observed, and only representative results related to condition A∗ are presentedand discussed here. For this condition, the combustor wall-temperature initially starts from473.4. Transient behavior of the pressure oscillations for warm-start conditionsFig. 3.5. (a) Combustor wall-temperature variation with time. (b) Spectrogram of the pressureoscillations. (c) PDF of the pressure oscillations. (d, g, and j), (e, h, and k), and (f, i, and l)pertain to [0.1 3], [201 205], and [484 500] s, respectively.680 K and plateaus at about 790 K, with the corresponding variation shown by the solid blackcurve in Fig. 3.5(a). The spectrogram of the pressure oscillations for condition A∗ is presentedin Fig. 3.5(b). The results in the figure show that, similar to the cold-start conditions, the483.4. Transient behavior of the pressure oscillations for warm-start conditionspressure oscillations feature three dominant modes (denoted by f1−3 in the figure). The ignitiontime occurs at about 0.1 s. The spectral analysis shows that, during 0.1 s  t  120 s, onlytwo modes of the oscillations (f1 and f3) are apparent. The results in Fig. 3.4(c) show that thepressure oscillations PDF features a combination of two bi-modal distributions, related to largeand small-amplitudes. As can be seen, the probability of occurrence of large (small)-amplitudeoscillations decrease (increase) as the combustor wall-temperature increases. A representativepressure signal related to t = [0.1 3] s is shown in Fig. 3.5(d). The corresponding phase-spacetrajectory and the Poincare´ map are presented in Figs. 3.5(g) and (j), respectively. The resultsin Figs. 3.5(b), (g), and (j) suggest that the pressure signal features limit cycle oscillations oncethe burner is ignited. During 120 s  t  300 s, all three modes of oscillations (f1−3) appear inthe spectrogram of the pressure oscillations, see Fig. 3.5(b). This is accompanied by weakeningof the large-amplitude limit cycle oscillations and appearance of small-amplitude oscillations.An example of these pressure oscillations is shown in Fig. 3.5(e) for t = [201 205] s. This typeof oscillations leads to a bi-orbital phase-space trajectory (Fig. 3.5(h)), as well as four clustersof points in the corresponding Poincare´ map (Fig. 3.5(k)).After t ≈ 300 s, the results in Fig. 3.5(b) show that the third mode of the pressure os-cillations disappears, but the first and the second modes dominate. During this time period,the probability of the occurrence of small-amplitude versus large-amplitude orbits increase, asshown in Fig. 3.5(c). A representative pressure signal related to t = [484 500] s is shown inFig. 3.5(f). As can be seen, the time duration related to small (large)-amplitude oscillationsincreases (decreases) compared to the results shown in Fig. 3.5(e). The phase-space trajectoryand the corresponding Poincare´ map related to t = [484 500] s are presented in Fig. 3.5(i) and(l), respectively. The results show that, after the combustor wall-temperature plateaus, the493.5. Relations between spatially averaged Helmholtz and Strouhal numbers versus the Reynolds numberpressure oscillations feature mixed-mode oscillations, which are different from the results pre-sented in Figs. 3.4(k) and (o) for cold-start conditions. Comparison of the results presented inFigs. 3.4 and 3.5 suggests that the warm-start conditions feature pressure oscillations which aremostly smaller amplitude compared to cold-start conditions. This is a desirable characteristicfrom a technical stand point and suggests the warm-start conditions could lead to improvedthermoacoustic behavior of the investigated small-scale power generation system.3.5 Relations between spatially averaged Helmholtz andStrouhal numbers versus the Reynolds numberPast studies [2, 26, 30, 57, 58] suggest that the Helmholtz and the Strouhal numbers remainunchanged for systems whose dynamics are dominated by the acoustics and the vortex-shedding,respectively. Such systems usually feature fixed combustor wall-temperature, and the Reynoldsnumber is changed by varying the mean bulk flow velocity. In the present setup, the combustionproducts exchange heat with the reactants and the water-cooled cap of the combustor. SeeFig. 2.1(b). This leads to spatial variation of the temperature inside the preheating/premixingsection as well as the combustion chamber. Thus, in addition to the temporal variation, thegas temperature varies spatially. These temporal and spatial temperature changes lead to thechange of the pressure oscillations frequency, gas kinematic viscosity, and the speed of sound,which cause variations of the Helmholtz, Strouhal, and Reynolds numbers. In this subsection,effects of such variations on the relations between the Helmholtz and Strouhal numbers with theReynolds number are investigated.Since the gas temperature varies throughout the preheating/premixing section, the combus-tion chamber, and the exhaust tube, the gas kinematic viscosity and the speed of sound were503.5. Relations between spatially averaged Helmholtz and Strouhal numbers versus the Reynolds numberaveraged spatially. Specifically, an axis (η) along the main flow stream is considered and shownin Fig. 3.6(a). In the figure, l1 − l8 are geometrical distances, and their values are provided inTable 3.1.0 100 256 400 500 600 700 800 900 10004006008000 385 440 600 26000500100015002225Fig. 3.6. (a) The experimental schematic along with η vector, (b) Temperature spatial variationfor η vector (c) Variation of wall-temperature and mixture temperature versus time for fuel-airequivalence ratio of 1Table 3.1. The distances in Fig. 3.6(a). The values are in millimetres.l1 l2 l3 l4 l5 l6 l7 l8 l9 l1065 280 40 55 100 35 85 60 600 1280The Reynolds and Helmholtz numbers were estimated based on the spatially averaged gaskinematic viscosity (ν) and the speed of sound (c), respectively, which are given byν =∫ lr0 νdηlr, (3.5a)513.5. Relations between spatially averaged Helmholtz and Strouhal numbers versus the Reynolds numberc =∫ l0 cdηl, (3.5b)where the preheating/premixing (lr) and the total (l) lengths are given by lr = Σ3i=1li and l =Σ10i=1li, respectively. Only the length of the preheating/premixing section is used for calculatingν, since the Reynolds number is estimated for the reactants. Compared to ν, c is estimatedbased on the temperature variation along the complete η-axis. Estimating the left-hand-sidesof Eqs. (3.5a) and (3.5b) requires information regarding variation of the temperature along theη-axis. In the present investigation, only the combustor wall (Tw), the preheated reactants(Tr), and the exhaust (Te) temperatures were measured. The thermocouple locations used formeasurement of Tw, Tr, and Te are shown along the η-axis in Fig. 3.6(b). Here Te is about300 K for all tested conditions. The cold reactants temperature (Tc) is assumed to be equalto Tr before ignition. This is because the heat transfer from the reactants to the exhaust tubeis negligible before ignition. For all tested conditions, Tc is equal to 290 K. It is assumed thatthe gas temperature reaches to the adiabatic flame temperature (Tad) at 55 mm downstream ofthe injection plane. This distance is referred to as l4 in Fig. 3.6(a). Sensitivity analysis showsthat changing l4 from about 0 to 155 (l4 + l5) does not significantly influence the Reynoldsnumber as well as the spatially averaged speed of sound. For all tested conditions, Tad wasestimated using Cantera [115] simulations of a freely propagating laminar methane-air flamewith fuel-air equivalence ratio set to that of the corresponding tested conditions. Among theabove five temperatures, only Tr and Tw vary with time, with variations measured and shown inFig. 3.6(c) for a representative tested condition (condition G) and at t = 256 s. Using the aboveargument and assuming a linear variation between Tc, Tr, Tad, Tw, and Te, the time variationof the spatially averaged gas kinematic viscosity and the speed of sound were estimated. Notethat the temperature decrease for the intervals of η = [440 600] mm and η = [600 2600] mm (in523.5. Relations between spatially averaged Helmholtz and Strouhal numbers versus the Reynolds numberFig. 3.6(b)) is due to the combustion products heat loss to the water-cooled combustion chamber(in Fig. 3.6(a)) and the cold reactants.Variations of the estimated Helmholtz and Strouhal numbers versus the Reynolds numberare presented in Figs. 3.7(a-c) and (d-f), respectively. The results in the first, second, and thethird columns pertain to the first, second, and the third modes of the pressure oscillations,respectively. Note that some modes of the pressure oscillations may be absent at different timeperiods. For example, for condition F and at Re ≈ 1800, the third mode of the oscillations isabsent, and as a result, no data point is shown for this Reynolds number and for condition F .00.511.522.500.51 1.522.500.511.522.52000 3000 400000.10.20.30.40.52000 3000 400000.10.20.30.40.52000 3000 400000.10.20.30.40.5Fig. 3.7. Variations of the Helmholtz (first row) and Strouhal (second row) numbers versus theReynold number. The first, second, and the third columns pertain to the first, second, and thethird modes, respectively.The results in Fig. 3.7 show that, for all tested conditions except E∗ and F , both variations ofHe−Re and St−Re nearly collapse. However, the results related to the experimental conditionsof E∗ and F do not collapse, and the values of He and St related to these two conditions aresmaller than those related to the rest of the tested experimental conditions. The results related533.5. Relations between spatially averaged Helmholtz and Strouhal numbers versus the Reynolds numberto E∗ and F conditions are highlighted by the dashed-ellipses in the figure. Nevertheless,for all tested conditions, the relation between the Strouhal number and the Reynolds numbernearly features a decreasing trend, which is similar to that presented in [2, 26, 57, 58]. However,compared to the results presented in these investigations [2, 26, 57, 58], the Helmholtz numberof our power generation system decreases with increasing the Reynolds number (decreasingtemperature). This is because decreasing the temperature decreases the spatially averagedspeed of sound and, as a result, the Helmholtz number. The above argument suggests that thedynamics of the thermoacoustic oscillations of our power generation system with details shownin Fig. 2.1 is dominated by neither the acoustics nor the vortex-shedding.54Chapter 4Contributions of structural, acoustic,and intrinsic modes to Rayleigh gainUsing the pressure and flame chemiluminescence data, thermoacoustic modes are identified,and their contributions to the Rayleigh gain are investigated in this chapter. The results pre-sented here pertain to the tested conditions shown in Fig. 2.6 as well as Tables 2.2 and 2.3. Allflame shapes are fixed for the presented conditions here. As discussed in Appendix A, flamesrelated to condition D2 can transition from Bunsen to M-shaped flames. Here, the analysis isperformed for the Bunsen-shaped flame of D2 condition.4.1 Characterization of structural and acoustic oscillationsIn order to identify the combustor natural structural mode, with a dominant frequency of fs,as well as the fundamental acoustic mode, with a dominant frequency of fa, mechanical impulsetests were performed.The exhaust structural displacement along x, y, and z directions, as well as air pressurefluctuations signals were recorded, and representative results are presented in Figs. 4.1(a-c) and(d) using the solid blue, red, green, and brown curves, respectively. In the figures, representa-tive displacement and pressure data associated with only two impulses are presented; however,554.1. Characterization of structural and acoustic oscillations0 0.2 0.4 0.6 0.8-0.200.20 0.5 1 1.5 2-0.1500.150 0.5 1 1.5 2-0.200.20 0.5 1 1.5 2-0.1500.1521 61 100 150 200 25000.0090.180.0270.0360.1 0.15 0.2-0.0100.015 5.3-.0150.015Fig. 4.1. (a-f) are samples of the exhaust displacement (along three axes) due to mechanicalimpulse, pressure oscillations of gas inside the setup excited by the mechanical impulse, pressureoscillations generated by speaker excitation at 200 Hz, and vibrations produced by the speakerexcitations at 90 Hz, respectively. (g) is the power spectrum densities of the results presentedin (a-f).data related to 10 impulses were collected and analyzed. Specifically, the signals are dividedinto 10 segments (corresponding to 10 impulses), the power spectrum density associated witheach segment is obtained, and the averaged normalized power spectrum density (PSD∗) arerespectively presented by the blue, red, green, and brown curves in Fig. 4.1(g). The results inFig. 4.1(g) are stepped for clarity. As can be seen, the PSD∗ of the displacement signals (blue,red, and green curves in Fig. 4.1(g)) feature a dominant peak around 21 Hz which is the naturalstructural frequency. The pressure oscillations PSD∗ (brown curve in Fig. 4.1(g)) features adominant peak near 61 Hz which is the fundamental acoustic frequency of the setup.In order to assess the capability of the pressure transducer for capturing the dominantfrequencies, additional experiments were performed. Specifically, a Zylux acoustic speaker waspositioned near the exhaust tube outlet and operated at an excitation frequency of 200 Hz, withthe corresponding pressure data shown in Fig. 4.1(e). The PSD∗ of the collected pressure data564.2. Pressure and spatially averaged flame chemiluminescence oscillations characteristicswas obtained and presented by the cyan colour in Fig. 4.1(g). A similar excitation experimentwas performed to assess the accelerometer reading, but for a different frequency of 90 Hz. Forthis experiment, the displacement and the corresponding PSD∗ are shown by the pink colourcurves in Figs. 4.1 (f) and (g), respectively. The results pertaining to both displacement andpressure oscillations suggest dominant peaks at the excitation frequencies. The above discussionsshow that the utilized accelerometer and pressure transducer can allow for reliable and accuratemeasurements of the structural displacement and pressure dominant frequencies. Here fs =21 Hz and fac = 61 Hz are the natural structural and fundamental acoustic frequencies of thesetup, respectively.4.2 Pressure and spatially averaged flame chemiluminescenceoscillations characteristicsIn this subsection, spectral analysis is performed to study the dominant modes of the pressure(for non-reacting) as well as pressure and flame chemiluminescence (for reacting) data.4.2.1 Non-reacting flow dominant modesPower spectrum density of the non-reacting pressure oscillations are presented in Fig. 4.2.For the results presented in Fig. 4.2, the mixture of fuel and air flow through the combustor, butthe mixture is not ignited. Results presented in the first and second columns pertain to fuel-airequivalence ratios of 0.7 and 0.9, respectively. The results presented in the first and second rowsare associated with air volumetric flowrates of 120 and 180, respectively. Green, black, red,and blue curves pertain to the igniter rod position of −5, 1.5, 2.5, and 5 mm, respectively. Twodominant frequencies exist in the pressure PSDs, fs = 21 Hz and fac = 61 Hz. These frequencies574.2. Pressure and spatially averaged flame chemiluminescence oscillations characteristicsare shown by the dashed-lines in the figures for clarity purposes. The results in Fig. 4.2 showthat these dominant frequencies do not change by changing the fuel-air equivalence ratio, airvolumetric flowrate, and location of the igniter rod. As discussed earlier, the modes near 21 and61 Hz correspond to the structural and acoustic modes of the setup, respectively.0 100 200 300 400 5001000200030000 100 200 300 400 500Fig. 4.2. Power spectrum density of the pressure oscillations for non-reacting conditions. S isthe location of the igniter rod.4.2.2 Reacting flow dominant modesPower spectrum density of the pressure and spatially averaged flame chemiluminescenceoscillations for fuel-air equivalence ratio of 0.7 and 0.9 are presented in Figs. 4.3 and 4.4, re-spectively. In both figures, the first and second rows pertain to PSD of the pressure andflame chemiluminescence data, respectively. The first and second columns correspond to airvolumetric flowrates of 120 and 180 SLPM, respectively. In each figure, green, black, red, andblue colours highlight the igniter rod position of −5, 1.5, 2.5, and 5 mm, respectively. Solidand dashed-curves pertain to Bunsen and M-shaped flames, respectively. As can be seen, thepressure oscillations PSD (Figs. 4.3(a and b)) features two dominant peaks near the naturalstructural (fs) and fundamental acoustic (fa) modes, with the latter being more pronounced.Compared to the pressure oscillations PSD, that of the flame chemiluminescence oscillations is584.2. Pressure and spatially averaged flame chemiluminescence oscillations characteristicsrelatively large near the structure natural frequency compared to the acoustic frequency. Theresults show that increasing the air volumetric flowrate does not change fa. This is becausethe flame chemiluminescence (heat release rate) occurs over a relatively small region (comparedto the size of the entire setup) and the overall gas temperature does not change substantiallycompared to non-reacting conditions. However, the value of the pressure oscillations PSD nearfa almost doubles by increasing Q˙air from 120 to 180 SLPM, which is similar to the observationsrelated to the fundamental acoustic modes reported in [26, 30, 40, 43, 44].0 100 200 300 400 500&&&Fig. 4.3. Power spectrum density of the pressure oscillations (first row) and flame chemilumines-cence (second row) oscillations for reacting conditions. The results pertain to fuel-air equivalenceratio of 0.7. Solid and dashed-curves highlight Bunsen and M-shaped flames, respectively.The PSDs of the pressure and flame chemiluminescence oscillations for fuel-air equivalenceratio of 0.9 are shown in Fig. 4.4. The first and second rows pertain to PSDs of the pressureand flame chemiluminescence. The first and second columns pertain to air volumetric flowratesof 120 and 180 SLPM. The solid and dashed-curves pertain to Bunsen and M-shaped flames,respectively. In addition to the dominant peaks evident in PSD of pressure oscillations (fs andfa) at φ = 0.7, increasing the fuel-air equivalence ratio to φ = 0.9 leads to the appearance oftwo new modes near 240 and 294 Hz as shown in Fig. 4.4. These two dominant frequencies donot correspond to either the structural or the acoustic modes and pertain to the first (ITA,1)594.2. Pressure and spatially averaged flame chemiluminescence oscillations characteristicsand second (ITA,2) intrinsic thermoacoustic modes. The results in Fig. 4.4 show, compared toPSDs of the pressure oscillations, those of the flame chemiluminescence feature dominant peaksnear fs, fITA,1, and fITA,2.0 100 200 300 400 500Fig. 4.4. Power spectrum density of the pressure (first row) and flame chemiluminescence (secondrow) oscillations at fuel-air equivalence ratio of 0.9. Solid curves pertain to Bunsen and dashed-curves pertain to M-shaped flames.For Q˙air = 120 SLPM, the results presented in Fig. 4.4(a and c) show that the thermoacousticoscillations are dominant near fITA,1 for the tested conditions that feature the M-shaped flameconfiguration. However, increasing Q˙air from 120 to 180 SLPM leads to appearance of dominantmodes near fs, fa, fITA,1, and fITA,2 for pressure as well as fs, fITA,1, and fITA,2 for flamechemiluminescence oscillations. As can be seen, increasing Q˙air increases the amplitude of thepressure oscillations PSD near fa, but the PSD values at fITA,1 and fITA,2 decrease. Also,these intrinsic thermoacoustic frequencies increase by increasing Q˙air, which is similar to theobservations reported in [39, 40]. Specifically, changing the air volumetric flowrate from 120to 180 SLPM increases fITA,1 from 238 to 250 Hz and fITA,2 from 287 to 300 Hz. The resultsshow that the PSDs near fITA,1 are more pronounced for M-shaped flames and at smaller Q˙air(compare the dashed- and the solid curves around fITA,1 in Figs. 4.4(a) and (c)). However,PSDs near fITA,2 are more pronounced for Bunsen flames, and the PSD amplitude increases by604.3. Spatial relevance of flame chemiluminescence to thermoacoustic modesincreasing air volumetric flowrate (compare the solid and dashed-curves near fITA,2 in Figs. 4.4(b)and (d)).4.3 Spatial relevance of flame chemiluminescence tothermoacoustic modesAlthough the results presented in the previous subsection allow for understanding the dom-inant thermoacoustic modes, it does not allow us to investigate which regions in the combustorpertain to the discussed modes. The analyses performed here allow us to understand the signifi-cance of flame chemiluminescence contribution associated with each mode inside the combustor.In order to investigate this, regions inside the combustion chamber (highlighted by bluecolour in Figs. 2.4(a) and (b)) that are oscillating dominantly around each mode are identified.To do this, first, the OH∗ signal is spatially averaged inside 3×3 windows. Then, the calculatedPSD associated with each window is separately partitioned around 5–30, 35–81, 220–260, and260–320 Hz, relating to fs, fa, fITA,1 and fITA,2, respectively. Finally, the values of PSD insideeach partition is averaged and referred to as P˜SD. For brevity purposes, P˜SD related to flamespertaining to minimum and maximum Q˙air are presented in Figs. 4.5 and 4.6, respectively. TheP˜SD related to fs and fa for Q˙air = 120 SLPM and φ = 0.7 are shown in the first and secondrows of Fig. 4.5, respectively. For φ = 0.9, P˜SD associated with fs, fa, and fITA,1 are shownin the third, fourth, and fifth rows of the figure, respectively. At 120 SLPM, Flames A1,3,4 andE1 are Bunsen, but Flames A2 and E2−4 are M-shaped. See Figs 2.6. Note that, for the richerBunsen flames, the fITA,1 does not exist, thus the corresponding P˜SD is marked as absent. Theresults in Fig. 4.5 show that large values of P˜SD related to fs are mostly associated with regionsclose the flame-holder (in Figs. 4.5(a, c, d, and i)) for Bunsen flames, and both regions close to614.3. Spatial relevance of flame chemiluminescence to thermoacoustic modesthe flame-holder and inner wings of the M-shaped flames (A2 and E2−4). See Figs. 4.5(b andj-l).Fig. 4.5. Flame chemiluminescence P˜SD for Q˙air = 120 SLPM. The first and second rowspertain to P˜SD evaluated near f = fs and f = fac for φ = 0.7, respectively. The third, fourth,and fifth rows pertain to P˜SD evaluated at f = fs, fa, and fITA,1, respectively. A1–A4 andE1–E4 are the tested conditions.For the data conditioned near f = fac, the M-shaped flames feature large values of P˜SD far-ther away from the flame-holder compared to the results associated with f = fs. In Figs. 4.5(f,n–p, and r–t), high intensity P˜SD regions whose P˜SD values are larger than 60% of the maxi-mum P˜SD value in the entire domain of investigation are highlighted by dashed-curves. For theleaner M-shaped flame (A2), since the flame is long (compared to E2−4), the high intensity P˜SDregions are elongated and are positioned closer to the stagnation wall. See the dashed-curves inFig. 4.5(f). However, for the richer M-shaped flames (E2−4), the high intensity regions are closerto the flame-holder, due to the flames being relatively shorter. The results show that movingthe igniter rod tip downstream (S changing from 1.5 to 5 mm) leads to the high intensity re-gions on the upper and lower tips of the M-shaped flames to become more symmetric. See thedashed-curves in Figs. 4.5(n–p). The values of P˜SD evaluated near fITA,1 (in Figs. 4.5(r–t)) are624.3. Spatial relevance of flame chemiluminescence to thermoacoustic modeslargest compared to the rest of the P˜SD values associated with other modes. See the maximumvalue of the colour bar in the last row. These large values of P˜SD pertain to Flames E2−4,which are M-shaped flames. The results show that moving the igniter rod tip downstream (Schanging from 1.5 to 5 mm) moves the high intensity regions evaluated near fITA,1 and positionsthem close to the main tube side walls. See dashed-curves in Figs. 4.5(r–t)).Fig. 4.6. Flame chemiluminescence P˜SD for Q˙air = 180 SLPM. The first and second rowspertain to P˜SD evaluated near f = fs and f = fac for φ = 0.7, respectively. The third, fourth,fifth, and sixth rows pertain to P˜SD evaluated at f = fs, fa, fITA,1, and fITA,2, respectively.D1–D4 and H1–H4 are the tested flames conditions.The P˜SD related to fs and fa for Q˙air = 180 SLPM and φ = 0.7 are shown in the first andsecond rows of Fig. 4.6, respectively. For the same air flowrates but at φ = 0.9, the third, fourth,fifth, and sixth rows pertain to P˜SD evaluated near fs, fa, fITA,1, and fITA,2, respectively. At180 SLPM of Q˙air, Flames D1−4 and H1 are Bunsen, and Flames H2−4 are M-shaped. SeeFigs 2.6. Note that for the richer Bunsen flames, fITA,1 does not exist, and the correspondingP˜SD is marked as absent. Comparison of the results presented in Figs. 4.5 and 4.6 suggest asQ˙air increases from 120 to 180 SLPM, some of the high intensity P˜SD regions related to fs and634.4. Relations between the Rayleigh gain and pressure oscillations amplitudefa move downstream. For example, P˜SD related to fs moves toward the stagnation wall, as seenby comparing the results in Figs. 4.5(j–l) with those in Figs. 4.6(j–l). In Figs. 4.6(n–p), highintensity P˜SD regions whose P˜SD values are larger than 60% of the maximum P˜SD value inthe entire domain of investigation are highlighted by dashed-curves. As Q˙air increases, the highintensity P˜SD pertaining to M-shaped flames evaluated at fa moves towards the stagnation wall,as seen by comparing the dashed-curves tip locations in Figs. 4.5(n–p) with those in Figs. 4.6(n–p). Comparison of the results in Figs. 4.5(r–t) with those in Figs. 4.6(r–t) suggest that theflame chemiluminescence related to the first intrinsic thermoacoustic mode primarily pertains toregions close to the stagnation wall. The P˜SD related to the M-shaped flames second intrinsicmode (Figs. 4.6(v–x)) features large values in regions similar to those of the first intrinsic mode,which are regions near the stagnation wall. However, P˜SD related to the Bunsen flame secondintrinsic mode features large values near both quartz tube inner wall and stagnation wall, asshown in Fig. 4.6(u).4.4 Relations between the Rayleigh gain and pressureoscillations amplitudeIn order to estimate the Rayleigh gain associated with each of the thermoacoustic modes,first, the local flame chemiluminescence data, along with the pressure data were band-passfiltered around 5–30, 35–81, 220–260, and 260–320 Hz to estimated the Rayleigh gain related tofs, fa, fITA,1, and fITA,2, respectively. Then, the band-pass filtered data along with Eq. (1.2)was used to estimate the local Rayleigh gain. Finally, the Rayleigh gain was averaged overseveral cycles of oscillations estimating the time-averaged Rayleigh gain (ψ). This subsectionaims to study the contributions of structural, acoustic, and intrinsic thermoacoustic modes to644.4. Relations between the Rayleigh gain and pressure oscillations amplitudethe spatially and temporally averaged Rayleigh gain (ψ). Effects of air volumetric flowrate andigniter rod location on the contributions are studied here. The results presented in Figs. 4.3and 4.4 suggested that the appearance of the thermoacoustic modes is significantly dependenton the tested fuel-air equivalence ratio. Thus, the spatially and temporally averaged Rayleighgain is separately presented for φ = 0.7 and φ = 0.9 in Figs. 4.7 and 4.8, respectively. In thefigures, the star and solid circular data symbols correspond to M-shaped and Bunsen flames,respectively. Green, black, red, and blue colour curves correspond to igniter tip locations of −5,1.5, 2.5, and 5 mm, respectively.120 140 160 18000.00040.00080.000120.00016120 140 160 18000.0050.0010.0015120 140 160 18000.00070.00140.00210.002840 50 60 70 80-0.00100.0010.0020.003Fig. 4.7. (a) and (b) are the spatially and temporally averaged Rayleigh gain for the thermoa-coustic oscillations band-pass filtered around f = fs and fa, respectively. (c) is the summationof results in (a) and (b). (d) is the variation of the pressure fluctuations root mean square, and(e) is variation of the spatially and temporally averaged Rayleigh gain versus p′rms. The resultspertain to φ = 0.7.Since, for φ = 0.7, the oscillations are dominated by the structural and acoustic modes, thespatially and temporally averaged Rayleigh gain is presented only for these modes in Figs. 4.7(a)and (b), respectively. The summation of the contributions related to both of these modes are654.4. Relations between the Rayleigh gain and pressure oscillations amplitudeshown in Fig. 4.7(c). As can be seen, increasing the air volumetric flowrate, generally, increasesψ associated with both modes as well as the total spatially and temporally averaged Rayleighgain. Results in Fig. 4.7(d) show that increasing Q˙air (and as result ψ) also, generally, increasesthe pressure oscillations root mean square (p′rms). As a result, ψ and p′rms are positively relatedfor φ = 0.7, as shown in Fig. 4.7(e). Similar to the results presented in Fig. 4.7, contributions ofconstituent modes to ψ were estimated for φ = 0.9, and the results pertaining to the structural,acoustic, and both intrinsic thermoacoustic modes are presented in Figs. 4.8(a–d), respectively.The summation of the Rayleigh gain contributions are also shown in Fig. 4.8(e).120 140 160 18000.00050.0010.0015120 140 160 18000.00050.0010.0015120 140 160 180-0.000700.0007120 140 160 180-0.00100.001120 140 160 18000.00070.00140.00210.002835 40 45 50 55 60 65-0.00100.0010.0020.003Fig. 4.8. (a), (b), (c), and (d) are the spatially and temporally averaged Rayleigh gain for thethermoacoustic oscillations band-pass filtered around f = fs, f = fac, f = fITA,1, and fITA,2,respectively. (e) presents summation of results in (a–d). (f) is the variation of the pressurefluctuations root mean square, and (g) is the variation of the spatially and temporally averagedRayleigh gain versus p′rms. The results pertain to φ = 0.9.As shown in Fig. 4.8, similar to the results associated with φ = 0.7 presented in Fig. 4.7, ψrelated to the structural and acoustic modes is mostly positive; and, as a result, these modesare thermoacoustically undesired. The results show that for φ = 0.9, increasing Q˙air, generally,decreases ψ associated with both intrinsic modes. Especially, at Q˙air of 160 and 180 SLPM, thevalues of ψ related to the intrinsic modes can become negative. This is important as it suggests664.5. Critical regions related to the structural, acoustic, and intrinsic thermoacoustic modesthat positive Rayleigh gain generated by the acoustic and structural modes can be mitigatedby the intrinsic thermoacoustic modes. Compared to the results associated with φ = 0.7, thoseassociated with φ = 0.9 show that increasing the air volumetric flowrate generally decreasesthe total ψ, as shown in Fig. 4.8(e). This is due to the contribution of intrinsic thermoacousticmodes. The variation of p′rms with Q˙air and ψ are presented in Figs. 4.8(f) and (g), respectively.Comparison of the results presented in Fig. 4.7(d) and Fig. 4.8(f) shows that p′rms correlationwith Q˙air decreases with increasing the fuel-air equivalence ratio. Comparison of the results inFig. 4.7 with those in Fig. 4.8 shows that increasing φ decreases p′rms as well as the spatially andtemporally averaged Rayleigh gain due to appearance of the intrinsic thermoacoustic modes.4.5 Critical regions related to the structural, acoustic, andintrinsic thermoacoustic modesIn order to study the critical regions that contribute to large p′rms presented in Section4.4, the time-averaged Rayleigh gain (ψ) was estimated for volumetric air flowrates of 120 and180 SLPM, and the results are presented in Figs. 4.9 and 4.10, respectively. The first intrinsicthermoacoustic mode is not present for Flames E1 and H1. Thus, the corresponding results aremarked as absent in Figs. 4.9 and 4.10. In Fig. 4.9, ψ related to fs and fa for Q˙air = 120 SLPMand φ = 0.7 are shown in the first and second rows, respectively. For the same air flowrate andat φ = 0.9, the third, fourth, and fifth rows pertain to ψ associated with fs, fa, and fITA,1,respectively. In Fig. 4.10, ψ related to fs and fa for Q˙air = 180 SLPM and φ = 0.7 are shownin the first and second rows, respectively. The third, fourth, fifth, and sixth rows pertain to ψassociated with fs, fa, fITA,1, and fITA,2, respectively, for φ = 0.9.As shown in Figs. 4.9(a–d and i–l) and 4.10(a–d and i–l), ψ related to fs features positive674.5. Critical regions related to the structural, acoustic, and intrinsic thermoacoustic modesFig. 4.9. The first and second rows are ψ related to fs and fa for φ = 0.7, respectively. Thethird, fourth, and fifth rows pertain to ψ evaluated near fs, fa, and fITA,1, respectively, forφ = 0.9. The results pertain to Q˙air = 120 SLPM.Fig. 4.10. The first and second rows are ψ evaluated near fs and fa for φ = 0.7, respectively.The third, fourth, fifth, and sixth rows pertain to ψ evaluated near fs, fa, fITA,1, and fITA,2,respectively, for φ = 0.9. The results pertain to Q˙air = 180 SLPM.values near the flame-holder, for both tested air volumetric flowrates and fuel-air equivalenceratios. For Bunsen flames at φ = 0.7 and Q˙air = 120 SLPM, ψ related to fs feature negativevalues near the stagnation wall, see Figs. 4.9(a), (c), and (d). At φ = 0.9 and for S = −5684.5. Critical regions related to the structural, acoustic, and intrinsic thermoacoustic modesand 1.5 mm, the Rayleigh gain is nearly positive in the domain of investigation for both testedvolumetric air flowrates. See Figs. 4.9(i and j) and 4.10 (i and j). While ψ is positive in thedomain of investigation for φ = 0.9 and S = 2.5 and 5 mm, the regions with positive Rayleighgain values become less pronounce by increasing the air volumetric flowrate for these igniter rodpositions, as shown in Figs. 4.10(k and l).Values of ψ¯ associated with fa depend on the flame configuration and the tested condition.For Bunsen flames, ψ¯ is generally positive as shown in Figs. 4.9(e, g, h, and m) and Figs. 4.10(e–h and m). However, for M-shaped flames, ψ¯ is positive at the central V-flame, and it canbecome negative farther downstream and for y > 0. See Figs. 4.9(f and n–p) and Figs. 4.10(n–p). Moving the igniter rod tip downstream (S changing from 1.5 to 5 mm) leads to the abovenegative regions to become less pronounced. See Figs. 4.10(o and p). Decreasing φ from 0.9 to0.7 and/or increasing Q˙air from 120 to 180 SLPM elongates both the positive and negative ψ¯regions towards the stagnation wall for both flame configurations.Time-averaged Rayleigh gain related to fITA,1 features strong axial variations, as shown inFigs. 4.9(r–t) and Figs. 4.10(r–t). Generally, for fITA,1, ψ¯ features negative Rayleigh gain nearthe flame-holder. Increasing the air volumetric flowrate from 120 to 180 SLPM increases the sizeand intensity of the regions that feature positive/negative ψ¯. Compare the results in Figs. 4.9(r–t) and Figs. 4.10(r–t). For Q˙air = 120 SLPM, ψ¯ is primarily positive near the stagnation wall,but it becomes negative as the volume flowrate increases to 180 SLPM. Similar to the firstintrinsic thermoacoustic mode, Rayleigh gain associated with the second intrinsic mode alsofeatures strong axial variations. Generally, the absolute values of the averaged Rayleigh gainestimated around the first intrinsic thermoacoustic mode are larger than those of the secondintrinsic thermoacoustic mode as evident in Figs. 4.10(u–x).694.5. Critical regions related to the structural, acoustic, and intrinsic thermoacoustic modesIn essence, the results presented in Figs. 4.9 and 4.10 suggest that regions that feature bothpositive and negative Rayleigh gain exist for all dominant modes, which is similar to the findingsreported in the literature. See for example [13–17]. Our results show that, generally, the absolutevalue of ψ¯ is relatively large for the intrinsic modes compared to the structural and acousticmodes. When φ is increased and the intrinsic modes appear, ψ¯ associated with these modesbecome strongly negative at the exit of the burner and near the flame-holder canceling out thepositive contribution of ψ¯ associated with fs and fa, and as a result, decreasing the overallψ¯. It is concluded that, although regions near the burner exist plane (or the flame-holder) cancontribute positively to ψ¯ at f = fs and fa, they can feature negative ψ¯ at f = fITA,1 and fITA,2.This observation suggests appearance of the intrinsic thermoacoustic modes can be helpful formitigation of thermoacoustic oscillations amplitude, and is reported here for the first time tothe best knowledge of the author.70Chapter 5ConclusionsConcluding remarks of the presented study and future directions are discussed in the firstand second subsections, respectively.5.1 Concluding remarksCharacteristics of thermoacoustic oscillations inside the combustion chamber of a small-scale power generation unit were investigated using two experimental setups. The first setupwas nearly an identical replication of that used in the engineering application. The second setupwas the simplified version of the first one, but it featured optical accessibility. Both versionsoperated under the mode of turbulent premixed combustion, and methane was used as the fuel.The results generated utilizing the first setup (Fig. 2.1) were used to investigate the influenceof the transient combustor wall-temperature on the thermoacoustic oscillations of the small-scale power generator during cold- and warm-start conditions. The fuel-air equivalence ratioand the volumetric air flowrate were varied from 0.7 to 1.4 and 50 to 130 standard liters perminute, respectively. For all tested conditions, the combustor wall-temperature increased overtime. One pressure transducer and three thermocouples were used to simultaneously record thepressure and temperature (at the combustor wall, warm reactants, and exhaust), respectively, ata frequency of 10 kHz. Long-term spectral analysis of the pressure oscillations suggests that the715.1. Concluding remarksoscillations feature three dominant modes. The frequencies of these modes are nearly insensitiveto the mean bulk flow velocity; however, increasing the fuel-air equivalence ratio increases thefrequencies. For the cold-start conditions, the results show that the pressure oscillations feature achaotic behavior initially. However, as the combustor wall-temperature increases, the oscillationstransition from featuring chaotic behavior to featuring limit cycle, chaotic, bursting, and thenagain limit cycle behavior. Compared to the cold-start conditions, the warm-start conditions areassociated with a different transition in the pressure oscillations. Specifically, for the warm-startconditions, the pressure signal features a combination of large- and small-amplitude limit cycleoscillations. The results show that, for these conditions, as the combustor wall-temperatureincreases, the probability of occurrence of the large-amplitude oscillations decreases and thatfor small-amplitude oscillations increases. This is of importance as the warm-start conditionsmay be preferred for the operation of the power generation unit. For both cold- and warm-startconditions, the spatially averaged values of the speed of sound and gas kinematic viscosity wereused to estimate the Helmholtz and the Strouhal numbers, respectively. The results show thatincreasing the Reynolds number decreases both the Helmholtz and Strouhal numbers. Thissuggests the dynamics of the thermoacoustic oscillations in the engineering-related device isdominated by neither the acoustics nor the vortex-shedding.The second experimental setup, whose combustion chamber has optical accessibility, was usedto investigate contributions of natural structural, acoustic, and intrinsic thermoacoustic modesto Rayleigh gain. In total, 32 conditions associated with fuel-air equivalence ratios of 0.7 and 0.9,four igniter rod locations of −5, 1.5, 2.5, and 5 mm, and four air volumetric flowrates of 120, 140,160, and 180 SLPM were examined. Synchronized pressure and OH∗ chemiluminescence datawere acquired at frequencies of 50 and 5 kHz, respectively. The results show that, depending on725.1. Concluding remarksthe tested condition, either Bunsen or M-shaped flames can be formed. The pressure oscillations(related to reacting conditions) may feature three types of thermoacoustic modes depending onthe tested fuel-air equivalence ratio. Specifically, the power spectrum densities related to flameswith φ = 0.7 feature dominant peaks at the natural structural and acoustic modes around 21and 61 Hz, respectively. However, power spectrum densities related to flames with φ = 0.9feature natural structural, acoustic, as well as two intrinsic thermoacoustic modes (near 240and 290 Hz). Spatially averaged Rayleigh gain associated with the structural and acousticmodes contribute positively to Rayleigh gain. Generally, this positive contribution increases byincreasing the air volumetric flowrate. As a result, pressure oscillations root mean square of theflames with fuel air-equivalence ratio of 0.7 increases with increasing air volumetric flowrate.Although intrinsic thermoacoustic modes have positive contributions for relatively small airvolumetric flowrates (120 SLPM), increasing this parameter decreases the values of the spatiallyaveraged Rayleigh gain for these modes. This is linked to the spatial variation of the Rayleighgain associated with the intrinsic thermoacoustic modes. Specifically, the Rayleigh gain fieldsrelated to these modes feature positive (negative) values near the stagnation wall for small (large)volumetric air flowrates. The negative Rayleigh gain near the stagnation wall leads to negativevalues of spatially averaged Rayleigh gain for large volumetric air flowrates. Thus, existence ofintrinsic thermoacoustic modes allows for decreasing the spatially averaged Rayleigh gain, whichis desirable for the operation of the combustor.Power spectral density of pressure oscillations for flames with fuel-air equivalence ratio of 0.9in both setups feature low frequency modes (below 70 Hz) and high frequency modes (beyond200 Hz). Although high frequency modes almost match in both setups, the low frequencymodes behave differently. This is linked to the simplification made in the optically accessible735.2. Future directionssetup design, which are removal of preheating and modification of the utilized flame-holder.5.2 Future directionsFour main directions for future investigations are recommended. First, the results presentedin this thesis pertain to a limited range of volumetric air flowrates. For a fixed fuel-air equiva-lence ratio, increasing the volumetric air flowrate increases the generated power, which is desired.However, our understanding related to characteristics of thermoacoustic oscillations as well asthe flame existence in the exhaust tubes is limited. Future experiments can include larger airvolumetric flowrates. Second, the utilized fuel is methane. This is normally tested to replicatenatural gas used in engineering applications. However, natural gas contains a few percentagesof hydrogen. This percentage may increase in future fuels used in Canada for decarbonizationpurposes. Effects of hydrogen addition on the reported thermoacoustic oscillations are of inter-est and are not studied here. Third, the second experimental setup features improved unburnthydrocarbon emissions, based on measurements that are not reported in the thesis. 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Open sounce combustion insta-bility low order simulator (oscilos–long) technical report.90Appendix AThermoacoustic oscillationscharacteristics during flame shapetransitionResults presented in Chapter 4 pertain to operation of the burner immediately after igni-tion. For all tested conditions, the flame shapes were presented in Fig. 2.6. Amongst all testedconditions, only D2 can feature a flame shape transition during the burner operation. In thisAppendix, first, characteristics of the thermoacoustic oscillations during D2 flame shape tran-sition are investigated. Then, it is shown that the flame shape transition can influence flameexistence in the exhaust tubes (shown by red colour in Figs. 2.4(a and b)). The flame existencein the exhaust tubes is detrimental to the operation of the burner, and its characteristics areinvestigated in the second subsection of this Appendix.A.1 Temporal and spectral thermoacoustic characteristicsduring flame shape transitionTemporal and spectral characteristics of Flame D2 shape during transition is discussed here.To elaborate this transition, the variation of the spatially averaged flame chemiluminescence91A.1. Temporal and spectral thermoacoustic characteristics during flame shape transitioninside the main tube during about 5 s operation of the burner and 1.5 s after ignition is presentedin Fig. A.1(a). For clarity purposes, the sliding mean of the results in Fig. A.1(a) was estimatedusing a window size of 0.2 s, and the results are shown by the solid blue curve.1.5 2 2.5 3 3.4 3.8 4 4.5 5 5.5 6 6.51.72.22.71.5 2 2.5 3 3.4 3.8 4 4.5 5 5.5 6 6.5-0.2500.250 25 50 75 850 25 50 75 85-13013Fig. A.1. (a) and (d) pertain to spatially averaged flame chemiluminescence in the main tubeand pressure oscillations related to Flame D2 condition, respectively. (b) and (c) are OH∗chemiluminescence at t = 3.09 and 4.09 s, respectively.As shown in the shaded gray area, for t = [3.4 3.8] s, the spatially averaged flame chemi-luminescence increases, which is evident by about 14% increase in the sliding averaged data.Representative flame chemiluminescence before and after this increase corresponding to t = 3.09and t = 4.09 s are shown in Figs. A.1(b) and (c), respectively. The results in Figs. A.1(b) and (c)show that the flame shape transitions from Bunsen to M-shaped for Flame D2 condition. Thepressure oscillations corresponding to results in Fig. A.1(a) are presented in Fig. A.1(d). As canbe seen, although the mean (averaged over 0.2 s) spatially averaged flame chemiluminescenceincrease during flame shape transition, the pressure oscillations do not seem to be influenced by92A.1. Temporal and spectral thermoacoustic characteristics during flame shape transitionthis transition. In order to further investigate the flame shape transition, the pressure oscilla-tions spectrogram is obtained and presented in Fig. A.2(a). Also, the power spectrum densitiesof the pressure oscillations pertaining Flame D2 and associated with the Bunsen and M-shapedflames are shown by solid and dashed-black curves in Fig. A.2(b), respectively. The power spec-1.5 3.4 3.8 112161150Fig. A.2. (a) and (b) are the pressure oscillations spectrogram and power spectrum density forFlame D2 condition, respectively.trum densities of Flame D2 are conditioned based on the flame shape for the result presented inFig. A.2(b). The result presented in Fig. A.2(a) show that the pressure oscillations spectrogramfeatures a strong peak near fa (acoustic frequency) and a relatively weak peak near fs (structuralfrequency). As can be seen, the spectrogram does not feature a noticeable change near the timeassociated with flame shape transition (see red highlighted region in Fig. A.2(a)). However, theoscillations around fa become stronger as the flame shape transitions from Bunsen to M-shaped.93A.2. Characteristics of main and exhaust tubes flame chemiluminescenceIn agreement with this, power spectrum density of D2 is about 14% larger for the time periodthat it features M-shaped than the time period that D2 features Bunsen flame, as can be seenin Fig. A.2(b).A.2 Characteristics of main and exhaust tubes flamechemiluminescenceThe flame shape transition in addition to thermoacoustic oscillations can influence the exis-tence of flame in the exhaust tubes. Please note, regardless of the tested conditions, the flamescan exist in the exhaust tubes. In this subsection, Flames A1–D1 and A2–D2 are utilized tostudy the effect of tested conditions on the amount of flame chemiluminescence in the main andexhaust tubes.Variations of the spatially averaged flame chemiluminescence in the main (OH∗M (t)) andexhaust (OH∗E (t)) tubes are shown in Figs. A.3(a) and (b), respectively. The results pertain toFlame D1 condition. Although the spatially averaged chemiluminescence signal inside the maintube features an oscillatory behavior, the spatially averaged chemiluminescence signal inside theexhaust tube does not feature an oscillatory behavior.Variation of the spatially and temporally averaged flame chemiluminescence inside the main,〈OH∗〉M, and one of the exhaust tubes,〈OH∗〉E, are presented in Figs. A.4(a) and (b), respec-tively. Error bars associated with〈OH∗〉Mare overlaid in Fig. A.4(a) and are estimated as onestandard deviation of OH∗M (t) signals. In both figures, blue hollow and solid red circular datapoints pertain to Bunsen and M-shaped flames, respectively. In both figures, two data pointsare shown for condition D2 as it transitions from Bunsen to M-shaped. Note that the value of〈OH∗E〉is estimated for one exhaust tube. Taking into account the flame chemiluminescence94A.2. Characteristics of main and exhaust tubes flame chemiluminescence1.41.61.822.21.1 1.2 1.3 1.4 1.500.050.10.150.21.1 1.12 1.1400.050.1Fig. A.3. (a) and (b) are the spatially averaged OH∗ chemiluminescence signal inside the mainand one exhaust tube, respectively. The results pertain to Flame D1 condition.in all exhaust tubes, it can be seen that the total flame chemiluminescence in all exhaust tubesis significant compared to〈OH∗M〉.BunsenM-shapeddata1data2Fig. A.4. (a) and (b) are the spatially and temporally averaged flame chemiluminescence in themain and exhaust tubes, respectively.The results in the Fig. A.4 show that increasing Q˙air increases the averaged flame chemilu-minescence inside both main and exhaust tubes for both Bunsen and M-shaped flame configura-95A.2. Characteristics of main and exhaust tubes flame chemiluminescencetions. It can be seen that the flames in the main tube feature relatively larger chemiluminescencefor M-shaped flames compared to Bunsen flames. However, the flame chemiluminescence in theexhaust tubes is larger for Bunsen flames than the M-shaped flames. In essence, the resultspresented in this Appendix show that flames can exist in the exhaust tubes and their chemi-luminescence increase moving from M-shaped to Bunsen flames as well as increasing the airvolumetric flowrate. As a result, operation conditions that promote M-shaped flames formationand smaller Q˙air are recommended for avoiding formation of flames in the exhaust tubes.96
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Experimental study of thermoacoustic oscillations inside a small-scale power generator combustor
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