Data assimilation in a nonlinear time-delayed dynamical system

When the heat released by a flame is sufficiently in phase with the acoustic pressure, a self-excited thermoacoustic oscillation can arise. These nonlinear oscillations are one of the biggest challenges faced in the design of safe and reliable gas turbines and rocket motors. In the worst-case scenario, uncontrolled thermoacoustic oscillations can shake an engine apart. Reduced-order thermoacoustic models, which are nonlinear and time-delayed, can only qualitatively predict thermoacoustic oscillations. To make reduced-order models quantitatively predictive, we develop a data assimilation framework for state estimation. We numerically estimate the most likely nonlinear state of a Galerkin-discretized time delayed model of a horizontal Rijke tube, which is a prototypical combustor. Data assimilation is an optimal blending of observations with previous state estimates (background) to produce optimal initial conditions. A cost functional is defined to measure the statistical distance between the model output and the measurements from experiments; and the distance between the initial conditions and the background knowledge. Its minimum corresponds to the optimal state, which is computed by Lagrangian optimization with the aid of adjoint equations. We study the influence of the number of Galerkin modes, which are the natural acoustic modes of the duct, with which the model is discretized. We show that decomposing the measured pressure signal in a finite number of modes is an effective way to enhance state estimation, especially when nonlinear modal interactions occur during the assimilation window. This work represents the first application of data assimilation to nonlinear thermoacoustics, which opens up new possibilities for real-time calibration of reduced-order models with experimental measurements.