FKPP fronts in cellular flows: the large-P\'eclet regime

We investigate the propagation of chemical fronts arising in Fisher--Kolmogorov--Petrovskii--Piskunov (FKPP) type models in the presence of a steady cellular flow. In the long-time limit, a steadily propagating pulsating front is established. Its speed, on which we focus, can be obtained by solving an eigenvalue problem closely related to large-deviation theory. We employ asymptotic methods to solve this eigenvalue problem in the limit of small molecular diffusivity (large P\'eclet number, $\text{Pe} \gg 1$) and arbitrary reaction rate (arbitrary Damk\"ohler number $\text{Da}$). We identify three regimes corresponding to the distinguished limits $\text{Da} = O(\text{Pe}^{-1})$, $\text{Da}=O\left((\log \text{Pe})^{-1}\right)$ and $\text{Da} = O(\text{Pe})$ and, in each regime, obtain the front speed in terms of a different non-trivial function of the relevant combination of $\text{Pe}$ and $\text{Da}$. Closed-form expressions for the speed, characterised by power-law and logarithmic dependences on $\text{Da}$ and $\text{Pe}$ and valid in intermediate regimes, are deduced as limiting cases. Taken together, our asymptotic results provide a complete description of the complex dependence of the front speed on $\text{Da}$ for $\text{Pe} \gg 1$. They are confirmed by numerical solutions of the eigenvalue problem determining the front speed, and illustrated by a number of numerical simulations of the advection--diffusion--reaction equation.