Modelling and control of combustion instabilities with anchored laminar ducted flames

196 Pages·2015·10.32 MB·English

Imperial College London
Department of Aeronautics
Modelling and control of combustion instabilities with
anchored laminar ducted flames
Charles Michael Angelo Luzzato
August 2, 2015
Supervised by Dr. Aimee Morgans
Submitted in part fulfilment of the requirements for the degree of
Doctor of Philosophy in Aeronautics of Imperial College London
and the Diploma of the Imperial College
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Abstract
This thesis deals with the derivation of new semi-analytical methods for the modelling of
combustion instabilities in anchored laminar flame combustors. In a first part, through an
analysis of the motion of the acoustic discontinuity in a ducted flame model, it shows that
the movement of the flame induced discontinuity can lead to stability changes. For unstable
combustors, it can also affect the amplitude of limit cycle oscillations. In a second part, the
problems that are encountered when attempting to obtain the transfer functions for linearly
unstablesystemsfromwithinlimitcyclearedemonstrated. Indeed,underthesecircumstances,
both the phase and amplitude of the unstable mode need to be corrected. Whilst the correc-
tion to the phase can easily be determined, the correction to the gain cannot, supporting the
need for robust model based controllers or adaptive control methods which do not require sys-
tem identification. Lastly, this thesis presents the derivation and implementation of the first
asymptotic-based mathematical models which account for the flame motion, hydrodynamic
field and acoustic field in an anchored ducted flame setup. This modelling exploits the differ-
ence in length scales associated with the flame, hydrodynamic field and acoustic waves. Unlike
ducted flame models which omit the hydrodynamic field, this allows us to capture instability
mechanisms such as Rayleigh-Taylor, or Darrieus-Landau instabilities, in the context of an-
chored laminar flames. This is done for two simplified configurations: a weakly conical flame
shape, and a conical flame shape case with small mean heat release.
Key words: G-Equation, lean-premixed flame, combustion instabilities, thermo-acoustic in-
stability,Rayleighcriterion,hydrodynamicinstability,transferfunctionfromlimitcycle,robust
control, active adaptive control, Darrieus-Landau instability, parametric instability, V-flame,
anchored flame, time-varying acoustics
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Declaration and Copyright
I herewith certify that all material in this dissertation which is not my own work has been
properly acknowledged.
Imperial College, London Charles Michael Angelo Luzzato
The copyright of this thesis rests with the author and is made available under a Creative
Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy,
distribute or transmit the thesis on the condition that they attribute it, that they do not use
it for commercial purposes and that they do not alter, transform or build upon it. For any
reuse or redistribution, researchers must make clear to others the licence terms of this work.
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Contents
Acronyms 13
List of Symbols 15
1 Introduction 21
1.1 Lean premixed combustors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.2 Combustion instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.2.1 The Rayleigh criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Pressure and heat release coupling . . . . . . . . . . . . . . . . . . . . . . 27
Saturation into limit cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.2.2 Models for combustion instabilities . . . . . . . . . . . . . . . . . . . . . . 30
Flame modelling and the G-Equation . . . . . . . . . . . . . . . . . . . . 31
Acoustics modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Modelling in the presence of hydrodynamics . . . . . . . . . . . . . . . . . 35
1.3 The need for prediction or control . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.4 Objectives and summary of research . . . . . . . . . . . . . . . . . . . . . . . . . 37
2 Acoustic wave modelling: accounting for time variation in the compact
flame assumption 39
2.1 Plane wave acoustic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.1.1 Method of characteristics / wave description method . . . . . . . . . . . . 41
Invariants and wave expansion . . . . . . . . . . . . . . . . . . . . . . . . 41
Implementation in the case of a ducted flame model . . . . . . . . . . . . 41
Boundary condition and perturbations . . . . . . . . . . . . . . . . . . . . 43
Acoustic jumps across the flame . . . . . . . . . . . . . . . . . . . . . . . 44
2.1.2 Accuracy of the numerical implementation of the method of characteristics 45
Comparison between linear, polynomial, and spline interpolation . . . . . 45
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2.1.3 The importance of mean flow . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2 Demonstrating the importance of time-variation . . . . . . . . . . . . . . . . . . 48
2.2.1 Floquet theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.3 Introducing time-variation in a ducted flame model . . . . . . . . . . . . . . . . 49
2.3.1 Movement of the flame . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.4 Effect of time-variation on combustion instabilities . . . . . . . . . . . . . . . . 51
2.4.1 Combustor stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4.2 Strongly unstable systems and limit cycle amplitude . . . . . . . . . . . . 55
2.4.3 Effect of the Rayleigh criterion . . . . . . . . . . . . . . . . . . . . . . . . 56
2.5 Acoustics modelling conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3 Active control of combustion instabilities 61
3.1 Obtaining linear models for model-based control from within limit cycle . . . . 62
3.1.1 Ducted flame limit cycle system . . . . . . . . . . . . . . . . . . . . . . . 62
3.1.2 Open loop transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1.3 Transfer function of a controlled system . . . . . . . . . . . . . . . . . . . 66
Active feedback controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Correction of transfer function obtained from limit cycle . . . . . . . . . . 68
3.2 Adaptive control of combustion instabilities . . . . . . . . . . . . . . . . . . . . 70
3.2.1 Background theory for Self Tuning Regulator adaptive controllers . . . . 71
State-space formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Controller coefficient update rules . . . . . . . . . . . . . . . . . . . . . . 72
3.2.2 Application to limit cycle systems with moving acoustic discontinuity . . 75
3.3 Transfer function analysis and control conclusions . . . . . . . . . . . . . . . . . 78
4 Flame modelling in the presence of hydrodynamics: theory 79
4.1 Modelling approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2 Derivation of the non-linear equations . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2.1 Non-dimensionalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2.2 Switching to the “flame” frame of reference . . . . . . . . . . . . . . . . . 83
4.2.3 Acoustic region: Linearised acoustics . . . . . . . . . . . . . . . . . . . . . 84
4.2.4 Hydrodynamic region: incompressible Euler equations . . . . . . . . . . . 86
4.2.5 Flame region: jump equations and flame equation . . . . . . . . . . . . . 87
Jumps across the flame . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Flame Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
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4.2.6 Flame anchoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Comparison of the G-Equation flame anchoring to the recirculation veloc-
ity method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3 Linearised hydrodynamics and flame . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3.1 Physical space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
The influence of the steady flame shape . . . . . . . . . . . . . . . . . . . 95
Weakly conical flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Conical flame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.3.2 Spectral space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Weakly conical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Conical flame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.4 Asymptotic expansion modelling conclusion . . . . . . . . . . . . . . . . . . . . 104
5 Flame modelling in the presence of hydrodynamics: numerical implemen-
tation and results 105
5.1 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.1.1 Implementation structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.1.2 Dealing with Fourier series of discontinuous functions . . . . . . . . . . . 106
Function continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Solving the continuation minimisation problem . . . . . . . . . . . . . . . 109
Function continuation for time marching methods . . . . . . . . . . . . . 111
5.1.3 Verification of the asymptotic model implementation . . . . . . . . . . . . 112
Acoustic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Numerical convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Signal extension method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.1.4 Validation of the weakly conical case . . . . . . . . . . . . . . . . . . . . . 115
5.2 Weakly conical results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2.1 Steady solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Amplitude of steady solution and linearisation assumptions . . . . . . . . 118
Mean position of the flame . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.2.2 The effect of the laminar burning velocity on anchoring . . . . . . . . . . 120
5.2.3 The effect of duct length . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2.4 Comparison to G-equation . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.3 Conical results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.3.1 Steady solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.3.2 Flame wrinkling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
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5.4 Conclusion on the implementation of asymptotic expansion modelling . . . . . . 131
6 Summary and Outlook 133
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.1.1 The importance of the location of the flame induced acoustic discontinuity 134
6.1.2 Designing plant based active controllers from limit cycle system data . . . 134
6.1.3 Modelling hydrodynamics for V-flames semi-analytically . . . . . . . . . . 135
6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.2.1 Short term research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.2.2 Other numerical methods: medium and long term research . . . . . . . . 137
Chebyshev points and polynomial interpolation: removing the Gibbs phe-
nomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Ghost fluid method: solving the non-linear hydrodynamic equations nu-
merically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Bibliography 143
Appendix 161
A Appendix: Acoustics of a ducted flame 161
A.1 Acoustic jump matrix for the G-equation . . . . . . . . . . . . . . . . . . . . . . 161
A.1.1 Steady acoustic jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
A.1.2 Fluctuating acoustic jumps . . . . . . . . . . . . . . . . . . . . . . . . . . 162
A.2 Analytical solution for the acoustics in a duct with heat release . . . . . . . . . 163
A.3 Time-variation: Mechanic-acoustic analogy . . . . . . . . . . . . . . . . . . . . . 165
A.3.1 Mechanical analogy results . . . . . . . . . . . . . . . . . . . . . . . . . . 166
A.3.2 Implications on Combustion Instabilities . . . . . . . . . . . . . . . . . . . 168
A.4 Time variation: Ritz-Rayleigh Method . . . . . . . . . . . . . . . . . . . . . . . 169
A.4.1 Derivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
A.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Time-invariant case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Time-varying case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
A.4.3 Implications on Combustion Instabilities . . . . . . . . . . . . . . . . . . . 175
B Appendix: Derivation of the update rules for S.T.R. adaptive control 177