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Research Papers

Stability and Limit Cycles of a Nonlinear Damper Acting on a Linearly Unstable Thermoacoustic Mode

[+] Author and Article Information
Claire Bourquard

CAPS Laboratory,
Department of Mechanical Engineering,
ETH Zürich,
ML J12.1, Sonneggstrasse 3,
Zürich 8092, Switzerland
e-mail: clairebo@ethz.ch

Nicolas Noiray

CAPS Laboratory,
Department of Mechanical Engineering,
ETH Zürich,
ML J12.1, Sonneggstrasse 3,
Zürich 8092, Switzerland
e-mail: noirayn@ethz.ch

Manuscript received November 9, 2018; final manuscript received November 14, 2018; published online December 17, 2018. Editor: Jerzy T. Sawicki.

J. Eng. Gas Turbines Power 141(5), 051012 (Dec 17, 2018) (8 pages) Paper No: GTP-18-1687; doi: 10.1115/1.4042080 History: Received November 09, 2018; Revised November 14, 2018

The resonant coupling between flames and acoustics is a growing issue for gas turbine manufacturers, which can be reduced by adding acoustic dampers on the combustion chamber walls. Nonetheless, if the engine is operated out of the stable window, the damper is exposed to high-amplitude acoustic levels, which trigger unwanted nonlinear effects. This work provides an overview of the dynamics of this coupled system using a simple analytical model, where a perfectly tuned damper is coupled to the combustion chamber. The damper, crossed by a purge flow in order to prevent hot gas ingestion, is modeled as a nonlinearly damped harmonic oscillator. The combustion chamber featuring a linearly unstable thermoacoustic mode is modeled as a Van der Pol oscillator. Analyzing the averaged amplitude equations gives the limit cycle amplitudes as function of the growth rate of the unstable mode and the mean velocity through the damper neck. Experiments are also performed on a simple rectangular cavity, where the thermoacoustic instability is mimicked by an electro-acoustic instability. A feedback loop is built, through which the growth rate of the instability can be controlled. A Helmholtz damper is added to the cavity and tuned to the mode of interest. The stabilization capabilities of the damper and the amplitude of the limit cycle in the unstable cases are in good agreement between the experiments and the analytical and numerical predictions, underlining the potentially dangerous behavior of the system, which should be taken into account for real engine cases.

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Figures

Grahic Jump Location
Fig. 1

Sketch of a Helmholtz resonator with dimensions. For simplicity the neck length is simply labeled “l” but the model includes an end correction on both sides of the neck.

Grahic Jump Location
Fig. 2

(a) Cavity limit cycle amplitude A as function of ν and u¯ for κ = −0.06 (s−1 Pa−2) (other parameters are the same as in the experimental setup introduced in the next section) coloring: gradient of the amplitude A. The transparent surface corresponds to the solutions of Eq. (25) (η and u not in phase), while the opaque surface corresponds to the solutions of Eq. (22) (η and u in phase) ((b), (c)) corresponding bifurcation diagram as function of ν for different fixed u¯ for the amplitude A (c) and the phase Δφ (b). Plain lines correspond to stable fixed points (i.e., limit cycles, thick =out-of-phase, thin = in phase), dotted lines correspond to unstable fixed points. (d) Corresponding bifurcation diagram as function of u¯ for different fixed ν. (e) Stability map as function of ν and u¯. Bold black lines: Hopf bifurcations corresponding to the linear model stability limit. Other bold lines: fold bifurcations merging in a cusp catastrophe point at (ν = 28.4 rad/s, u¯=2.31 m/s). Bottom left shaded area: limit-cycles corresponding to out-of-phase solution. Bottom right shaded area: zone of existence of an unstable fixed point. Top shaded area: zone where a single stable limit cycle exists. The area not shaded corresponds to the zone where the whole system is stable.

Grahic Jump Location
Fig. 3

(a) Sketch of the impedance tube used for the damper nonlinear model validation (b) cut of the Helmholtz damper (c) comparison between the resistive term obtained from a fit on the experimental reflection coefficient (triangles) and the resistive term obtained from the nonlinear model RNL (black line)

Grahic Jump Location
Fig. 4

Sketch of the measurement setup

Grahic Jump Location
Fig. 5

Measurement techniques: growth rate measurements (a) and growth rate ramping with effect on the pressure inside the cavity (b) and on the phase between the cavity pressure and the damper neck velocity (c) (which corresponds to π/2 subtracted to the phase between cavity pressure and damper volume pressure)

Grahic Jump Location
Fig. 6

Comparison between analytical results (a) and experiments (envelope of the pressure measured during ramping of ν,plain = ramping up, dashed = ramping down) (b). Black lines = cavity without damper. Besides the mismatch for the onset growth rate, one can see that the tendency is well reproduced, and all curves seem to collapse for higher growth rate even though their order is reversed.

Grahic Jump Location
Fig. 7

Comparison between numerical results from slowly ramping down the growth rate, highlighting the influence of detuning δ and of damper position with regard to mode shape Ψ: (a) u¯=1.3 m/s, (b) u¯=1.7 m/s, (c) u¯=2.1 m/s, and (d) u¯=2.5 m/s

Grahic Jump Location
Fig. 8

Comparison between simulink simulations and analytical model (left column) and experiments (right column). On the left, the results of the analytical model are superimposed on the simulink simulations, which highlights the perfect match between both. (a) and (b) represent the cavity without damper. Despite a different onset of instability, nonlinear features are well reproduced experimentally, for example, the jump in (d) and (f), and the hysteresis in (l) and (n).

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