# Oscillatory response of an idealized two-dimensional diffusion flame: Analytical and numerical study

Abstract

Abstract The heat release response of a two-dimensional (2-D) co-flow diffusion flame is theoretically investigated for the mass fraction fluctuations of reactant concentration at the inlet boundary and for time-varying spatially uniform flow velocity field in the domain. The present work is an extension of the Burke–Schumann steady flame model to an oscillatory situation, but in a 2-D framework. The governing equation of the problem is the scalar advection equation for the Schwab–Zel'dovich variable. An exact solution is found in terms of an infinite series for the case of mixture fraction fluctuations at the inlet boundary. The cases of time-varying uniform flow velocity field and a combination of velocity and mixture fraction fluctuations are investigated numerically. The temperature and heat release rate of the flame are thermodynamically calculated utilizing the mixed-is-burnt approach. The main results of the paper are the response functions. The nonlinearity in the calculation of the heat release rate and the convective nonlinearity due to velocity fluctuations result in the generation of higher harmonics in the response function for a given sinusoidal excitation. Therefore, the response function is decomposed and obtained for each of the significant harmonics. The results show that, in general, the response function decreases with increase in the excitation frequency as reported with premixed flames in the literature. However, in the present case, the decrease occurs when the excitation time-scale is less than the diffusion time-scale. Interesting flame shape variations such as flame clip-off, flipping between overexpanded and underexpanded conditions, and flame wrinkling are observed in the case of mixture fraction oscillations.