  We investigate the influence of steady periodic flows on the propagation of
chemical fronts in an infinite channel domain. We focus on the sharp front
arising in Fisher--Kolmogorov--Petrovskii--Piskunov (FKPP) type models in the
limit of small molecular diffusivity and fast reaction (large P\'eclet and
Damk\"ohler numbers, $\mathrm{Pe}$ and $\mathrm{Da}$) and on its heuristic
approximation by the G equation. We introduce a variational formulation that
expresses the two front speeds in terms of periodic trajectories minimizing the
time of travel across the period of the flow, under a constraint that differs
between the FKPP and G equations. This formulation shows that the FKPP front
speed is greater than or equal to the G equation front speed. We study the two
front speeds for a class of cellular vortex flows used in experiments. Using a
numerical implementation of the variational formulation, we show that the
differences between the two front speeds are modest for a broad range of
parameters. However, large differences appear when a strong mean flow opposes
front propagation; in particular, we identify a range of parameters for which
FKPP fronts can propagate against the flow while G fronts cannot. We verify our
computations against closed-form expressions derived for $\mathrm{Da}\ll
\mathrm{Pe}$ and for $\mathrm{Da}\gg \mathrm{Pe}$.
