Dispersion in the large-deviation regime. Part I: shear flows and periodic flows

摘要

The dispersion of a passive scalar in a fluid through the combined action ofadvection and molecular diffusion is often described as a diffusive process,with an effective diffusivity that is enhanced compared to the molecular value.However, this description fails to capture the tails of the scalarconcentration distribution in initial-value problems. To remedy this, wedevelop a large-deviation theory of scalar dispersion that provides anapproximation to the scalar concentration valid at much larger distances awayfrom the centre of mass, specifically distances that are $O(t)$ rather than$O(t^{1/2})$, where $t \gg 1$ is the time from the scalar release. The theorycentres on the calculation of a rate function obtained by solving aone-parameter family of eigenvalue problems which we derive using twoalternative approaches, one asymptotic, the other probabilistic. We emphasisethe connection between large deviations and homogenisation: a perturbativesolution of the eigenvalue problems reduces at leading order to the cellproblem of homogenisation theory. We consider two classes of flows in somedetail: shear flows and cellular flows. In both cases, large deviationgeneralises classical results on effective diffusivity and captures newphenomena relevant to the tails of the scalar distribution. These includeapproximately finite dispersion speeds arising at large P\'eclet number$\mathrm{Pe}$ (corresponding to small molecular diffusivity) and, fortwo-dimensional cellular flows, anisotropic dispersion. Explicit asymptoticresults are obtained for shear flows in the limit of large $\mathrm{Pe}$. (Acompanion paper, Part II, is devoted to the large-$\mathrm{Pe}$ asymptotictreatment of cellular flows.) The predictions of large-deviation theory arecompared with Monte Carlo simulations that estimate the tails of concentrationaccurately using importance sampling.