# 2017 - The effect of the time delay between pressure and heat release rate for low-order models of thermoacoustic instabilities

Abstract: A successful low-order model introduced by Schuermans et al. [19], Noiray et al. [13] studies thermoacoustic instabilities assuming the fluctuating heat release rate q to be in phase with the acoustic pressure p, by neglecting the component of q out of phase with p. In this investigation we remove this hypothesis and consider a model in which, if p peaks at time t, q will peak at a later time t + tau. We generalize the delay tau as the local slope of the flame phase response in the vicinity of the acoustic mode of interest with natural frequency w0. We present an alternative, simpler formulation of the problem and show that the low-order governing equations presented in [19,13] are actually the time derivative of it. We will first consider systems where two degenerate azimuthal modes oscillate, and then prove that most results apply also to systems where only one longitudinal mode oscillates. In the linear regime, we show that the system has a higher linear growth rate than the model where the part of q not in-phase with p is neglected. This effect is larger for larger values of the product tau w0, with w0 being the natural frequency of oscillation of the acoustic system. We also discuss how the local slope of the flame phase response plays a role in (de) stabilizing the thermoacoustic system. We then discuss in the nonlinear regime how to apply the method of averaging and the method of multiple timescales to this nonlinear problem, in particular accounting for the varying frequency of oscillation of the system, and validate the results with extensive numerical simulations. The resulting equations allow us to discuss the implications that a non-zero tau has on the capabilities of a successful method of growth-rate extraction of Noiray and Schuermans [15]. We present mathematical evidence suggesting that, within the limits of certain approximations, the method should identify a good estimate of the growth rate despite the simplified assumptions, in line with the past experience of Bothien et al. [2].