Fast chemical reaction in two-dimensional Navier-Stokes flow: Initial regime

Abstract

This paper studies an infinitely fast bimolecular chemical reaction in a two-dimensional bi-periodic Navier-Stokes flow. The reactants in stoichiometric quantities are initially segregated by infinite gradients. The focus is placed on the initial stage of the reaction characterized by a well-defined one dimensional material contact line between the reactants. Particular attention is given to the effect of the diffusion of the reactants. This study is an idealized framework for isentropic mixing in the lower stratosphere and is motivated by the need to better understand the effect of resolutionon stratospheric chemistry in Climate-Chemistry Models. Adopting a Lagrangian stretching theory approach, we relate theoretically the ensemble mean of the length of the contact line, of the gradients along it and of the modulus of the rate of decrease of the space averaged reactant concentrations (here called the chemical speed) to the joint statistics of the finite time Lyapunov exponent with two equivalent times. The inverse of the Lyapunov exponent measures the stretching time scale of a Lagrangian parcel on a chaotic orbit up to a finite time t, while the first equivalent time measures it in the recent past before t and the second equivalent time in the early part of the trajectory. We show that the chemical speed scales like the square root of the diffusion and that its time evolution is determined by rare large events in the finite time Lyapunov exponent distribution. The case of smooth initial gradients is also discussed. The theoretical results are tested with an ensemble of direct numerical simulations (DNS) using a pseudospectral model.