  This study investigates the linear stability of a laminar premixed flame,
anchored on a square cylinder and confined inside a channel. Many modern linear
analysis concepts have been developed and validated around non-reacting
bluff-body wake flows, and the objective of this paper is to explore whether
those tools can be applied with the same success to the study of reacting flows
in similar configurations. It is found that linear instability analysis of
steady reacting flow states accurately predicts critical flow parameters for
the onset of limit-cycle oscillations, when compared to direct numerical
simulation performed with a simple one-step reaction scheme in the low Mach
number limit. Furthermore, the linear analysis predicts a strong stabilising
effect of flame ignition, consistent with documented experiments and numerical
simulations. Instability in ignited wake flows is, however, found to set in at
sufficiently high Reynolds number, and a linear wavemaker analysis
characterises this instability as being driven by hydrodynamic mechanisms of a
similar nature as in non-reacting wake flows. The frequency of nonlinear
limit-cycle flame oscillations in this unstable regime is retrieved accurately
by linear eigenmode analysis performed on the time-averaged mean flow, under
the condition that the full set of the reacting flow equations is linearised.
If, on the contrary, unsteadiness in the density and in the reaction rate are
excluded from the linear model, then the congruence between linear and
nonlinear dynamics is lost.
