Title:

Passive scalar mixing in chaotic flows with boundaries

We are interested in examining the longtime decay rate of a passive scalar in twodimensional flows. The focus is on the effect of boundary conditions for kinematically prescribed velocity fields with random or periodic time dependence. Scalar evolution is followed numerically in a periodic geometry for families of flows that have either a slip or a noslip boundary condition on a square or plane layer subdomain D. The boundary conditions on the passive scalar are imposed on the boundary C of the domain D by restricting to a subclass invariant under certain symmetry transformations. The scalar field obeys constant (Dirichlet) or noflux (Neumann) conditions exactly for a flow with the slip boundary condition and approximately in the noslip case. At late times the decay of a passive scalar, for example temperature, is exponential in time with a decay rate gamma(kappa), where kappa is the molecular diffusivity. Scaling laws of the form gamma(kappa) ~ C*kappa^alpha for small kappa are obtained numerically for a variety of boundary conditions on flow and scalar, and supporting theoretical arguments are presented. In particular when the scalar field satisfies a Neumann condition on all boundaries, alpha ~ 0 for a slip flow condition; for a noslip condition we confirm results in the literature that alpha ~ 1/2 for a plane layer, but find alpha ~ 2/3 in a square subdomain D where the decay is controlled by stagnant flow in the corners. For cases where there is a Dirichlet boundary condition on one or more sides of the subdomain D, the exponent measuring the decay of the scalar field is alpha ~ 1/2 for a slip flow condition and alpha ~ 3/4 for a noslip condition. The scaling law exponents alpha for chaotic timeperiodic flows are compared with those for similarly constructed random flows. Motivated by the theory of passive scalar field, in Part II of this work we extend the investigation of the evolution of passive scalar for the flows addressed specifically in Part I. Based on an ensemble averaging over random velocity fields, the theoretical results obtained confirm the scaling laws computed numerically for a single, long realisation of random flows. In analogy with Lebedev and Turitsyn (2004) and Salman and Haynes (2007) our results show very good agreement between such an ensemble theory and applications. In part III of our study, we expand upon the work set out in the previous parts of this thesis in terms of the polarcoordinate system. We analyse the structures of flows driven near to a corner with a link to Moffatt corner eddies. A longtime exponential decay rate gamma(kappa)=C*kappa^alpha has been obtained confirming our numerical and theoretical results predicted in Part I and Part II in this work. The exponent alpha is determined in a structure of Moffatt corner eddies.
