Nonlinear self-excited oscillations of a ducted flame

Abstract

Self-excited oscillations of a confined flame, burning in the wake of a bluff-body flameholder, are considered. These oscillations occur due to interaction between unsteady combustion and acoustic waves. According to linear theory, flow disturbances grow exponentially with time. A theory for nonlinear oscillations is developed, exploiting the fact that the main nonlinearity is in the heat release rate, which essentially 'saturates'. The amplitudes of the pressure fluctuations are sufficiently small that the acoustic waves remain linear. The time evolution of the oscillations is determined by numerical integration and inclusion of nonlinear effects is found to lead to limit cycles of finite amplitude. The predicted limit cycles are compared with results from experiments and from linear theory. The amplitudes and spectra of the limit-cycle oscillations are in reasonable agreement with experiment. Linear theory is found to predict the frequency and mode shape of the nonlinear oscillations remarkably well. Moreover, we find that, for this type of nonlinearity, describing function analysis enables a good estimate of the limit-cycle amplitude to be obtained from linear theory.Active control has been successfully applied to eliminate these oscillations. We demonstrate the same effect by adding a feedback control system to our nonlinear model. This theory is used to explain why any linear controller capable of stabilizing the linear flow disturbances is also able to stabilize finite-amplitude oscillations in the nonlinear limit cycles.

On the thermo-acoustic Fant equation

“…Several measurements are taken from the experimental afterburner. Bloxsidge and Dowling empirically derive a flame transfer function for this configuration [31,42], having the form 47) where ω is the complex disturbance frequency, and τ 1 and τ 2 are the respective dynamic and convective time delay. τ 1 is not physically very well understood; however, τ 2 can be thought to be the time taken for the mean flow to convect across the combustion zone.…”

“…The combustor models used in our analysis, which are described in Chapter 3, are very similar to those used in [31,42,60] so we will use flame transfer function 2.47 in Chapters 7-9.…”

“…Dowling [31] describes a nonlinear heat release formulation introduced in Section 2.3 which predicts well when the heat release Q f (t) and downstream non-flame volume velocity Q(t) make only small-amplitude perturbations about a their mean or 'steady-state' valuesQ f andQ o , respectively. …”

“…Although interest in this type of source has recently revived [22][23][24][25][26][27][28][29][30], understanding factors governing feedback on the direct monopole flame source (i.e. aerodynamics of the oscillating mean flow, structural vibrations, flame flashback, flame-vortex interactions, burn rate fluctuations produced by flame-area oscillation, saturation of nonlinear heat release rates, etc) are probably more important for the control of the system instabilities [31][32][33][34][35][36][37].…”

“…Reduced complexity analyses that avoid a numerical treatment of the whole flow are generally based on one-dimensional acoustic models [31][32][33][42][43][44], where upstream and downstream acoustic waves are coupled to unsteady thermal and other sources in the combustion zone by application of appropriate conservation conditions. These analyses are limited because they do not account for aeroacoustic flow properties such as vortex shedding.…”