  We investigated a nonlinear advection-diffusion-reaction equation for a
passive scalar field. The purpose is to understand how the compressibility can
affect the front dynamics and the bulk burning rate. We study two classes of
flows: periodic shear flow and cellular flow both in the case of fast advection
regime, analysing the system at varying the extent of compressibility and the
reaction rate. We find that the bulk burning rate in a shear flow increases
with compressibility intensity.
  Furthermore, the faster the reaction the more important the difference with
respect to the laminar case. The effect has been quantitatively measured and it
turns out to be generally little. For the cellular flow, the two extreme cases
have been investigated, with the whole perturbation situated either in the
centre of the vortex or in the periphery. The dependence in this case does not
show a monotonic scaling with different behaviour in the two cases. The
enhancing remains modest and always less than 20%
