  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.
