  As lean premixed combustion systems are more susceptible to combustion
instabilities than non-premixed systems, there is an increasing demand for
improved numerical design tools that can predict the occurrence of combustion
instabilities with high accuracy. The inherent non-linearities in combustion
instabilities can be of crucial importance, and we here propose an approach in
which the one-dimensional Navier-Stokes and scalar transport equations are
solved for geometries of variable cross-section. The focus is on attached
flames, and for this purpose a new phenomenological model for the unsteady heat
release from a flame front is introduced. In the attached flame method (AFM)
the heat release occurs over the full length of the flame. The non-linear code
with the use of the AFM approach is validated against results from an
experimental study of thermoacoustic instabilities in oxy-fuel flames by
Ditaranto and Hals [Combustion and Flame, 146, 493-512 (2006)]. The numerical
simulations are in accordance with the experimental measurements and both the
frequencies and the amplitudes of the resonant acoustic pressure modes are
reproduced with good accuracy.
