A parsimonious system of ordinary differential equations for the response modeling of turbulent swirled flames

Abstract

In this work, we present a parsimonious set of ordinary differential equations (ODEs), describing with good precision and over a wide range of frequencies the linear and nonlinear dynamics of different types of fully premixed, turbulent, swirl-stabilized flames. This phenomenological model comprises eight ODEs and can physically be interpreted as a superposition of two mass–spring–damper systems.

The model is characterized by 11 parameters and one nonlinear term of structure $\beta x^2\dot{x}$ in particular, with position $x$ and rate of displacement $\dot{x}$ of a given oscillator mass. The model can be optimized in frequency domain or in time domain. The former is suitable for situations where experimental data of the flame transfer function (FTF) and flame describing function (FDF) are available. The latter is suitable for situations where time series (input and output) from numerical simulations or experiments are available. In the present work, large eddy simulation (LES) data is used.

Choosing a model comprising ODEs enables us to describe the dynamics of the flame using linear and nonlinear input-to-output transformations via the states. This is a sharp contrast to commonly applied finite impulse response (FIR) models, which can only directly employ nonlinear functions acting on the input or output — not on derivatives or other states of the underlying dynamical system. We demonstrate the simplicity of implementing our model by investigating the coupling of a time-domain acoustic solver with the set of ODEs proposed, all within the graphical interface of Simulink. The $x^2\dot{x}$ nonlinearity plays a crucial role in achieving good agreement in the nonlinear regime, as observed in the FDF, as well as in both the amplitude and frequency of the investigated limit cycle, which exhibits a simple period-one oscillation. However, there is still room for improvement in the nonlinear model if more complex dynamics need to be modeled, thanks to the flexible ODE structure proposed in this work.

Novelty and significance

A phenomenological model, characterized by a parsimonious set of ordinary differential equations is presented. This model is able to describe the dynamic flame response of typical turbulent, swirl-stabilized flames to acoustic disturbances in their linear and nonlinear regime. The model can be calibrated from broadband time series data or reference frequency domain data. After calibration, the model can be coupled to an acoustic network to predict the occurrence of limit cycles and (if they are present) their frequency as well as amplitude.