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Research Papers: Gas Turbines: Combustion, Fuels, and Emissions

# Nonlinear Phenomena in Thermoacoustic Systems With Premixed Flames

[+] Author and Article Information
Karthik Kashinath

Engineering Department,
University of Cambridge,
Cambridge CB2 1PZ, UK
e-mail: kk377@cam.ac.uk

Santosh Hemchandra

Institute of Aerodynamics,
RWTH Aachen University,
Aachen 52062, Germany
e-mail: s.hemchandra@aia.rwth-aachen.de

Matthew P. Juniper

Engineering Department,
University of Cambridge,
Cambridge CB2 1PZ, UK
e-mail: mpj1001@cam.ac.uk

1Corresponding author.

2Present address: Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India.

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the Journal of Engineering for Gas Turbines and Power. Manuscript received September 18, 2012; final manuscript received November 30, 2012; published online May 20, 2013. Editor: David Wisler.

J. Eng. Gas Turbines Power 135(6), 061502 (May 20, 2013) (9 pages) Paper No: GTP-12-1365; doi: 10.1115/1.4023305 History: Received September 18, 2012; Revised November 30, 2012

## Abstract

Nonlinear analysis of thermoacoustic instability is essential for the prediction of the frequencies, amplitudes, and stability of limit cycles. Limit cycles in thermoacoustic systems are reached when the energy input from driving processes and energy losses from damping processes balance each other over a cycle of the oscillation. In this paper, an integral relation for the rate of change of energy of a thermoacoustic system is derived. This relation is analogous to the well-known Rayleigh criterion in thermoacoustics, however, it can be used to calculate the amplitudes of limit cycles and their stability. The relation is applied to a thermoacoustic system of a ducted slot-stabilized 2-D premixed flame. The flame is modeled using a nonlinear kinematic model based on the $G$-equation, while the acoustics of planar waves in the tube are governed by linearized momentum and energy equations. Using open-loop forced simulations, the flame describing function (FDF) is calculated. The gain and phase information from the FDF is used with the integral relation to construct a cyclic integral rate of change of energy (CIRCE) diagram that indicates the amplitude and stability of limit cycles. This diagram is also used to identify the types of bifurcation the system exhibits and to find the minimum amplitude of excitation needed to reach a stable limit cycle from another linearly stable state for single-mode thermoacoustic systems. Furthermore, this diagram shows precisely how the choice of velocity model and the amplitude-dependence of the gain and the phase of the FDF influence the nonlinear dynamics of the system. Time domain simulations of the coupled thermoacoustic system are performed with a Galerkin discretization for acoustic pressure and velocity. Limit cycle calculations using a single mode, along with twenty modes, are compared against predictions from the CIRCE diagram. For the single mode system, the time domain calculations agree well with the frequency domain predictions. The heat release rate is highly nonlinear but, because there is only a single acoustic mode, this does not affect the limit cycle amplitude. For the twenty-mode system, however, the higher harmonics of the heat release rate and acoustic velocity interact, resulting in a larger limit cycle amplitude. Multimode simulations show that, in some situations, the contribution from higher harmonics to the nonlinear dynamics can be significant and must be considered for an accurate and comprehensive analysis of thermoacoustic systems.

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## Figures

Fig. 1

Schematic of the two-dimensional slot stabilized premixed flame in a duct: L0 is the length of the duct, x˜f is the flame position along the duct, α = 0.02 is the fraction of the duct cross-sectional area occupied by the burner and the flow is from left to right

Fig. 2

Instantaneous images of the flame during one forcing cycle; φ = 1.06, βf = 2.14, ɛ = 0.25, St = 1, and K = 2.5. The thick black line represents the slot burner and the thin black curve is the flame surface. Note the formation of sharp cusps towards the products, which is a distinct characteristic of premixed flames seen in experiments [26,27,35].

Fig. 3

Flame describing function FDF(ω,ɛ) = Q·˜'/Q·˜0/u˜'/u˜0: (a) gain, and (b) phase K = 2.5, βf = 2.14, φ = 1.06

Fig. 4

CIRCE diagram: driving (Ψdriv) – damping (Ψdamp) as a function of amplitude ɛ for thermoacoustic systems with different fundamental frequencies f*: (a) K = 1.0, (b) K = 1.5, and (c) K = 2.5

Fig. 5

CIRCE for thermoacoustic systems with different fundamental frequencies (duct lengths): K = 2.5, φ = 1.06, and βf = 2.14. (a) Slice of Fig. 4(c) at f* = 0.4: A is the stable limit cycle, and (b) slice of Fig. 4(c) at f* = 1.0; A is the unstable limit cycle, and B is the stable limit cycle.

Fig. 6

CIRCE for thermoacoustic systems with different fundamental frequencies (duct lengths): K = 2.5, φ = 1.06, and βf = 2.14. (a) Slice of Fig. 4(c) at f* = 1.2: A and C are the stable limit cycles, B is the unstable limit cycle, and (b) slice of Fig. 4(c) at f* = 1.8: A and C are the unstable limit cycles, B and D are the stable limit cycles.

Fig. 7

Amplitude-dependence of the fundamental of heat release rate oscillations and the phase between the heat release rate and velocity perturbations: K = 2.5, φ = 1.06, and βf = 2.14. (a) q1 at St = 1.0, (b) Δϕ1 at St = 1.0, and (c) sin(Δϕ1) at St = 1.0. The gray shaded area shows regions where driving is negative, while white shows where it is positive. (d) q1 at St = 0.8.

Fig. 8

Time domain calculations for self-excited thermoacoustic systems with one mode and twenty modes: f* = 0.87, K = 1.5, βf = 2.14, φ = 1.06, and ζ = 0.05(c1 = 0.03, c2 = 0.02). (a) Time trace of the acoustic velocity perturbations at the flame location xf, and (b) time trace of the contributions of the first five Galerkin modes to the acoustic velocity at the flame location xf of the 20-mode system.

Fig. 9

Nonlinear heat release rate oscillations in thermoacoustic systems with one mode and twenty modes: f* = 0.87, K = 1.5, βf = 2.14, φ = 1.06, and ζ = 0.05(c1 = 0.03, c2 = 0.02). (a) Time trace of the heat release rate (phase-shifted to avoid overlap of figures), and (b) Fourier transforms of the heat release rate.

Fig. 10

CIRCE diagram: the intersection of driving and damping curves represents the frequency domain prediction for the limit cycle amplitude. Time-domain limit cycle locations are marked by symbols: f* = 0.87, K = 1.5, βf = 2.14, φ = 1.06, and ζ = 0.05(c1 = 0.03, c2 = 0.02).

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