Diffusion and Reaction–Diffusion in Steady Flows at Large Péclet Numbers

Abstract

In many, if not most, geophysical flows molecular diffusion is formally negligible because its relative importance is measured by the inverse of a large Péclet number. This has motivated a number of studies, based in some way or another on various theoretical approaches to turbulence, where the assumed randomness of the turbulent flow plays the center role. If one adds to this complicated situation of the passive scalar the possibility of chemical reactions changing the chemical layout of the fluid, the number of unresolved issues increases dramatically. The chemistry is often such that, without flow, the reaction progresses by a front (the flame front in gaseous combustion) sweeping the system, across which the reaction takes place, from an unburnt (fresh) side to a burnt side. I look at these problems in the case of steady flows made of rolls, as generated by Rayleigh–Bénard instability for instance, certainly an idealization of real turbulence, although it is not clear by how much. The propagation of the chemical reaction in this system depends first on the diffusion of a passive scalar, an interesting question in the limit of a large Péclet number in a flow without open flow lines. The main result there is that the effective diffusion is somewhere in between the molecular diffusion and the “turbulent" diffusion. Once chemical reactions are taken into account, that is when one considers the reaction–diffusion case, one finds that the front speed is the laminar propagation velocity (without flow) times the Péclet number to the power 1/4. I refine this last result and give the behavior of the prefactor in the Zel’dovich limit of a narrow reaction zone.