A kinematic model of a ducted flame

Abstract

A premixed ducted flame, burning in the wake of a bluff-body flame-holder, is considered. For such a flame, interaction between acoustic waves and unsteady combustion can lead to self-excited oscillations. The concept of a time-invariant turbulent flame speed is used to develop a kinematic model of the response of the flame to flow disturbances. Variations in the oncoming flow velocity at the flame-holder drive perturbations in the flame initiation surface and hence in the instantaneous rate of heat release. For linear fluctuations, the transfer function between heat release and velocity can be determined analytically from the model and is in good agreement with experiment across a wide frequency range. For nonlinear fluctuations, the model reproduces the flame surface distortions seen in schlieren films.Coupling this kinematic flame model with an analysis of the acoustic waves generated in the duct by the unsteady combustion enables the time evolution of disturbances to be calculated. Self-excited oscillations occur above a critical fuel–air ratio. The frequency and amplitude of the resulting limit cycles are in satisfactory agreement with experiment. Flow reversal is predicted to occur during part of the limit-cycle oscillation and the flame then moves upstream of the flame-holder, just as in experimental visualizations. The main nonlinearity is identified in the rate of heat release, which essentially ‘saturates’ once the amplitude of the velocity fluctuation exceeds its mean. We show that, for this type of nonlinearity, describing function analysis can be used to give a good estimate of the limit-cycle frequency and amplitude from a quasi-nonlinear theory.

Combining a Helmholtz solver with the flame describing function to assess combustion instability in a premixed swirled combustor

“…It has been shown however that such interactions are of lesser importance to the combustion instability process (see for example Moeck et al (2012) where such interactions are investigated). When considering the FDF approach, the flame response depends not only on the frequency of the acoustic velocity perturbation, but also on its amplitude (Dowling, 1997(Dowling, , 1999Noiray et al, 2008;Durox et al, 2009). The FDF is then defined as…”

On the thermo-acoustic Fant equation

“…Additionally, turbulence, externally applied forces, and moving boundaries are neglected. We assert that the sound generated by turbulence in a real fluid is equivalent to that produced by an ideal, stationary acoustic medium forced by the stress distribution 32) where T ij is the Lighthill stress tensor, which is comprised of three terms. The first term ρv i v j is called the Reynolds stress; it is a nonlinear quantity due to random turbulent fluctuations in fluid momentum and can be neglected in the absence of turbulence.…”

“…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.…”

Nonlinear simulations of combustion instabilities with a quasi-1D Navier–Stokes code

“…A current practice in the modeling of unstable combustion systems is to apply a network model where the geometry of the combustor is modeled by a network of acoustic elements and a simplified form of the pressure equation is solved. The acoustic elements of a network model, also called multiports, correspond to various components of the system, e.g., the air or fuel supply, the transition between two regions of different cross-section, the outlet of the combustor, or the flame itself [6,7,8,9,10]. Mathematically, each multiport in the system is represented by a transfer function.…”