  We investigate self-excited axisymmetric oscillations of a lean premixed
methane--air V-flame in a laminar annular jet. The flame is anchored near the
rim of the centrebody, forming an inverted cone, while the strongest vorticity
is concentrated along the outer shear layer of the annular jet. Consequently,
the reaction and vorticity dynamics are largely separated, except where they
coalesce near the flame tip. The global eigenmodes corresponding to the
linearised reacting flow equations around the steady base state are computed in
an axisymmetric setting. We identify an arc branch of eigenmodes exhibiting
strong oscillations at the flame tip. The associated eigenvalues are robust
with respect to domain truncation and numerical discretisation, and they become
destabilised as the Reynolds number increases. The frequency of the leading
eigenmode is found to correspond to the Lagrangian disturbance advection time
from the nozzle outlet to the flame tip. Nonlinear time-resolved simulation
further reveals notable hysteresis phenomena in the subcritical regime prior to
instability. Hence, even when the flame is linearly stable, perturbations of
sufficient amplitude can trigger limit-cycle oscillations and
higher-dimensional dynamics sustained by nonlinear feedback. A Monte Carlo
simulation of passive tracers in the unsteady flame suggests a nonlinear
non-local instability mechanism. Notably, linear analysis of the subcritical
time-averaged limit-cycle state yields eigenvalues that do not match the
nonlinear periodic oscillation frequencies, attributed to the fundamentally
nonlinear dynamics of the subcritical V-flame instability, where the
dichromatic, non-local interaction cannot be approximated as a simple
distortion of the mean flow.
