  Previous studies have found that the flickering of buoyant diffusion flames
is associated with the periodic shedding of a toroidal vortex, which is formed
under gravity-induced shearing at the flames. Moreover, numerous experimental
investigations have reported the scaling relation, $f \propto D^{-1/2}$, where
$f$ is the flickering frequency and $D$ is the diameter of the fuel inlet.
However, the connection between the toroidal vortex mechanism and the scaling
relation has not been clearly understood. The current study theoretically
revisits this problem from the perspective of vortex dynamics. The theory
incorporates a recent finding in the vortex dynamics community: the detachment
of a continuously growing vortex ring is inevitable and can be dictated by a
universal constant that is essentially a non-dimensional circulation of the
vortex. By calculating the total circulation of the toroidal vortex and
applying the vortex ring detachment criterion, we mathematically established
the connection between the lifetime of a toroidal vortex and the flickering of
a buoyant diffusion flame, resulting in a general flickering frequency
formulation, which is validated by comparing with existing experimental data of
pool and jet diffusion flames.
