FKPP fronts in cellular flows: the large-Péclet 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 Peclet number, Pe≫1 ) and arbitrary reaction rate (arbitrary Damkohler number Da ).
We identify three regimes corresponding to the distinguished limits Da=O(Pe −1 ) , Da=O((logPe) −1 ) and Da=O(Pe) and, in each regime, obtain the front speed in terms of a different non-trivial function of the relevant combination of Pe and Da . Closed-form expressions for the speed, characterised by power-law and logarithmic dependences on Da and 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 Da for Pe≫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.