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

Abstract

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.

Stabilization of acoustic modes using Helmholtz and Quarter-Wave resonators tuned at exceptional points

“…Using a purely linear dissipation term is justified for the present study, which focuses on the linear stability limits. The reader can refer to [41] where the nonlinear problem is investigated. Eq.…”

“…In this context, the goal of the present experimental and analytical study is to build on the work proposed in [40] and investigate the potential of air-purged HH and QW dampers to increase modal damping in combustion chambers. This investigation focuses on the linear stability of the coupled system "dampers-combustion chamber", and the reader can refer to [41] for the complementary study dealing with the associated nonlinear dynamics. In the first part, the impedance of standalone HH and QW dampers is modeled and experimentally validated for a range of purge mass flows and damper volumes.…”

Effect of geometrical parameters and use of porous material in a Helmholtz resonator on suppression of thermo-acoustic instabilities

“…Out of all passive methods, Helmholtz Resonator has been widely used and researched. [16][17][18][19][20][21][22] Depending upon its shape and size a Helmholtz resonator can provide different amounts of damping. Helmholtz resonators can be subjected to a lot of modification to dampen the instabilities.…”