Chemical Front Propagation in Periodic Flows: FKPP versus G

Abstract

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 Péclet and Damköhler numbers, Pe and 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 makes it plain 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 Da Pe and for Da Pe.

Front propagation in cellular flows: scale dependence versus scale invariance

“…2003; Vladimirova et al. 2003; Tzella & Vanneste 2014, 2015, 2019), i.e. to propagation in a plane, they could not address the effect of the vortex structure on front propagation.…”

“…Numerical (Vladimirova et al. 2003) and theoretical (Tzella & Vanneste 2014, 2015, 2019) studies of the effective propagation provided by the ARD equation have been conducted in a 2-D space. Whereas they cannot address the effect of the overlooked transverse dimension and thus of the vortex structure, they may be relevant for the front propagation in a three-dimensional (3-D) fluid stirred by a planar flow, in particular regarding the implication of the vortex to front scale ratio in a vortex lattice.…”

“…2003; Paoletti & Solomon 2005; Pocheau & Harambat 2006, 2008; Mahoney et al. 2012; Tzella & Vanneste 2014, 2015, 2019; Beauvier, Bodea & Pocheau 2016, 2017). Then, as streamlines are confined to compact regions, no large-scale advection of the medium and thus no net kinematic effect are expected at long times on fluid particles.…”

“…2003; Paoletti & Solomon 2005; Pocheau & Harambat 2006, 2008; Mahoney et al. 2012; Tzella & Vanneste 2014, 2015, 2019; Beauvier et al. 2016, 2017).…”

“…We then determine the relationships between , and in both regimes and compare with numerical simulations (Vladimirova et al. 2003) and theoretical analysis (Tzella & Vanneste 2014, 2015, 2019) of ARD dynamics in a plane. Altogether, our study thus improves the characterization of front propagation in stirred flows and provides a detailed identification of a scale-invariant regime where spatial scales may be overlooked and of a scale-dependent regime where they significantly influence front propagation.…”